Limit inferior and limit superior

2178: 43: 191: 2908: 9822: 9297: 2712: 139:, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant. Limit inferior is also called 9536: 9011: 7185: 7006: 9594: 9069: 10578:

The "odd" and "even" elements of this sequence form two subsequences, ({0}, {1/2}, {2/3}, {3/4}, ...) and ({1}, {1/2}, {1/3}, {1/4}, ...), which have limit points 1 and 0, respectively, and so the outer or superior limit is the set {0,1} of these two points. However, there are no limit points that

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The "odd" and "even" elements of this sequence form two subsequences, ({0}, {0}, {0}, ...) and ({1}, {1}, {1}, ...), which have limit points 0 and 1, respectively, and so the outer or superior limit is the set {0,1} of these two points. However, there are no limit points that can be taken from the

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in that they bound a sequence only "in the limit"; the sequence may exceed the bound. However, with big-O notation the sequence can only exceed the bound in a finite prefix of the sequence, whereas the limit superior of a sequence like e may actually be less than all elements of the sequence. The

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of the real numbers to the real numbers. As in the case for sequences, the limit inferior and limit superior are always well-defined if we allow the values +∞ and −∞; in fact, if both agree then the limit exists and is equal to their common value (again possibly including the infinities). For

4952: 5089: 6794: 3510: 2903:{\displaystyle {\begin{alignedat}{4}\liminf _{n\to \infty }x_{n}&=\infty &&\;\;{\text{ implies }}\;\;\lim _{n\to \infty }x_{n}=\infty ,\\\limsup _{n\to \infty }x_{n}&=-\infty &&\;\;{\text{ implies }}\;\;\lim _{n\to \infty }x_{n}=-\infty .\end{alignedat}}} 1025: 750: 10744:. This circle, which is the Ω limit set of the system, is the outer limit of solution trajectories of the system. The circle represents the locus of a trajectory corresponding to a pure sinusoidal tone output; that is, the system output approaches/approximates a pure tone. 6650: 6415: 6272: 8426:

it contains the points and only the points which are in all except finitely many of the sets of the sequence. Since convergence in the discrete metric is the strictest form of convergence (i.e., requires the most), this definition of a limit set is the strictest possible.

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which crop up in analysis quite often. An interesting note is that this version subsumes the sequential version by considering sequences as functions from the natural numbers as a topological subspace of the extended real line, into the space (the closure of

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That is, the four elements that do not match the pattern do not affect the lim inf and lim sup because there are only finitely many of them. In fact, these elements could be placed anywhere in the sequence. So long as the

4818: 7012: 6833: 9817:{\displaystyle {\begin{aligned}\limsup _{n\to \infty }X_{n}&=\inf \,\{\,\sup \,\{X_{m}:m\in \{n,n+1,\ldots \}\}:n\in \{1,2,\dots \}\}\\&=\bigcap _{n=1}^{\infty }\left({\bigcup _{m=n}^{\infty }}X_{m}\right)\!.\end{aligned}}} 9292:{\displaystyle {\begin{aligned}\liminf _{n\to \infty }X_{n}&=\sup \,\{\,\inf \,\{X_{m}:m\in \{n,n+1,\ldots \}\}:n\in \{1,2,\dots \}\}\\&=\bigcup _{n=1}^{\infty }\left({\bigcap _{m=n}^{\infty }}X_{m}\right)\!.\end{aligned}}} 2389: 4957: 1628:

Whenever the ordinary limit exists, the limit inferior and limit superior are both equal to it; therefore, each can be considered a generalization of the ordinary limit which is primarily interesting in cases where the limit does

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only promise made is that some tail of the sequence can be bounded above by the limit superior plus an arbitrarily small positive constant, and bounded below by the limit inferior minus an arbitrarily small positive constant.

11754: 10914: 10840: 6655: 3385: 877: 602: 11377: 11198: 6514: 6279: 6136: 1723: 1191: 1109: 10740:) will oscillate endlessly after being perturbed (e.g., an ideal bell after being struck). Hence, if the position and velocity of this system are plotted against each other, trajectories will approach a circle in the 10938:

so that the suprema and infima always exist. In that case every set has a limit superior and a limit inferior. Also note that the limit inferior and the limit superior of a set do not have to be elements of the set.

2596: 12003: 9998: 5325: 4490: 4356: 11035: 11458: 11291: 4113: 5682: 8225: 1443: 1592: 9599: 9335: 9074: 8810: 1531: 1408: 5549: 3584: 5497: 5253: 774: 499: 3008: 2959: 11547: 8067: 7793: 4734: 4691: 5172: 1279: 386: 264: 2639: 5947: 5893: 2114: 1933: 9531:{\displaystyle {\begin{aligned}J_{n}&=\sup \,\{X_{m}:m\in \{n,n+1,n+2,\ldots \}\}\\&=\bigcup _{m=n}^{\infty }X_{m}=X_{n}\cup X_{n+1}\cup X_{n+2}\cup \cdots .\end{aligned}}} 9006:{\displaystyle {\begin{aligned}I_{n}&=\inf \,\{X_{m}:m\in \{n,n+1,n+2,\ldots \}\}\\&=\bigcap _{m=n}^{\infty }X_{m}=X_{n}\cap X_{n+1}\cap X_{n+2}\cap \cdots .\end{aligned}}} 8258:

does not include elements which are in all except finitely many complements of sets of the sequence. That is, this case specializes the general definition when the topology on set

3547: 11072: 6086: 7938: 7741: 7585: 2486: 11891: 6452: 5593: 11618: 4813: 4351: 4292: 5377: 5121: 4322: 3146: 2306: 8015: 7827: 7491: 5842: 5437: 2205: 2166: 1985: 8160: 8130: 2664: 11644: 6476: 2413: 2055: 1961:. In other words, any number larger than the limit superior is an eventual upper bound for the sequence. Only a finite number of elements of the sequence are greater than 1874: 11581: 10141: 8287: 7661: 4648: 3580: 2717: 11863: 8100: 5201: 3825: 3752: 3299: 3265: 5769: 4104: 4084: 4037: 4017: 3967: 3947: 3900: 3880: 3775: 3702: 2687: 11830: 7975: 7694: 7618: 7432: 2707: 2210: 1803: 1476: 1352: 1316: 1227: 381: 259: 11463:

If the limit inferior and limit superior agree, then there must be exactly one cluster point and the limit of the filter base is equal to this unique cluster point.

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whose relationship to limits of real-valued functions mirrors that of the relation between the limsup, liminf, and the limit of a real sequence. Take a metric space

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By this definition, a sequence of sets approaches a limiting set when the limiting set includes elements which are in all except finitely many sets of the sequence

7858: 7522: 5709: 4064: 3997: 3927: 3860: 3802: 3729: 3380: 3353: 3326: 3173: 2015: 1834: 10385: 1247: 7180:{\displaystyle \liminf _{x\to a}f(x)=\sup \,\{\,\inf \,\{f(x):x\in E\cap U\setminus \{a\}\}:U\ \mathrm {open} ,\,a\in U,\,E\cap U\setminus \{a\}\neq \emptyset \}} 7001:{\displaystyle \limsup _{x\to a}f(x)=\inf \,\{\,\sup \,\{f(x):x\in E\cap U\setminus \{a\}\}:U\ \mathrm {open} ,\,a\in U,\,E\cap U\setminus \{a\}\neq \emptyset \}} 2142:. In other words, any number below the limit inferior is an eventual lower bound for the sequence. Only a finite number of elements of the sequence are less than 7549: 7455: 5348: 3196: 12026: 11911: 11797: 11777: 11664: 11501: 7902: 7878: 7399: 6496: 6129: 6109: 6052: 6032: 6012: 5729: 4754: 3117: 2451: 2075: 2035: 1894: 1854: 7319:

of all of the accumulation sets. When ordering by set inclusion, the infimum limit is the greatest lower bound on the set of accumulation points because it is

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is the outer limit of solution trajectories of the system. Because trajectories become closer and closer to this limit set, the tails of these trajectories

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of all of the accumulation sets. When ordering by set inclusion, the supremum limit is the least upper bound on the set of accumulation points because it

11672: 7344:). Hence, when considering the convergence of a sequence of sets, it generally suffices to consider the convergence of the outer limit of that sequence. 10755:

The above definitions are inadequate for many technical applications. In fact, the definitions above are specializations of the following definitions.

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The following are several set convergence examples. They have been broken into sections with respect to the metric used to induce the topology on set

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around sets of points rather than single points themselves. That is, because each element of the sequence is itself a set, there exist accumulation

4947:{\displaystyle \limsup _{n\to \infty }\,(a_{n}b_{n})\leq \left(\limsup _{n\to \infty }a_{n}\!\right)\!\!\left(\limsup _{n\to \infty }b_{n}\!\right)} 1653: 1114: 1032: 216:

around the two limits. The superior limit is the larger of the two, and the inferior limit is the smaller. The inferior and superior limits agree

5084:{\displaystyle \liminf _{n\to \infty }\,(a_{n}b_{n})\geq \left(\liminf _{n\to \infty }a_{n}\right)\!\!\left(\liminf _{n\to \infty }b_{n}\right)} 12184: 12137: 5949:. The difference between the two is a rough measure of how "wildly" the function oscillates, and in observation of this fact, it is called the 5615: 8247:. Further discussion and examples from the set-theoretic point of view, as opposed to the topological point of view discussed below, are at 11919: 9883: 7401:

approaches a limiting set when the elements of each member of the sequence approach the elements of the limiting set. In particular, if

6789:{\displaystyle \liminf _{x\to a}f(x)=\sup _{\varepsilon >0}\left(\inf \,\{f(x):x\in E\cap B(a,\varepsilon )\setminus \{a\}\}\right).} 3505:{\displaystyle \liminf _{n\to \infty }x_{n}+\epsilon >x_{h_{n}}\;\;\;\;\;\;\;\;\;x_{k_{n}}>\limsup _{n\to \infty }x_{n}-\epsilon } 1020:{\displaystyle \limsup _{n\to \infty }x_{n}:=\inf _{n\geq 0}\,\sup _{m\geq n}x_{m}=\inf \,\{\,\sup \,\{\,x_{m}:m\geq n\,\}:n\geq 0\,\}.} 745:{\displaystyle \liminf _{n\to \infty }x_{n}:=\sup _{n\geq 0}\,\inf _{m\geq n}x_{m}=\sup \,\{\,\inf \,\{\,x_{m}:m\geq n\,\}:n\geq 0\,\}.} 6645:{\displaystyle \limsup _{x\to a}f(x)=\inf _{\varepsilon >0}\left(\sup \,\{f(x):x\in E\cap B(a,\varepsilon )\setminus \{a\}\}\right)} 6410:{\displaystyle \liminf _{x\to a}f(x)=\lim _{\varepsilon \to 0}\left(\inf \,\{f(x):x\in E\cap B(a,\varepsilon )\setminus \{a\}\}\right)} 6267:{\displaystyle \limsup _{x\to a}f(x)=\lim _{\varepsilon \to 0}\left(\sup \,\{f(x):x\in E\cap B(a,\varepsilon )\setminus \{a\}\}\right)} 5258: 5502: 5450: 8745:

that allows set intersection to generate a greatest lower bound and set union to generate a least upper bound. Thus, the infimum or

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would not be considered as convergence.) Since the limit inferior is at most the limit superior, the following conditions hold

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In each of these four cases, the elements of the limiting sets are not elements of any of the sets from the original sequence.

7246:, and so the supremum and infimum of any set of subsets (in terms of set inclusion) always exist. In particular, every subset 12160: 12113: 8165: 10362:, provide an important modification that "squashes" countably many (rather than just finitely many) interstitial additions. 1757: 1413: 7326:

Because ordering is by set inclusion, then the outer limit will always contain the inner limit (i.e., lim inf

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So the limit supremum is contained in all subsets which are upper bounds for all but finitely many sets of the sequence.

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Therefore, the limit inferior and limit superior of the sequence are equal to the limit superior and limit inferior of

4607:{\displaystyle \liminf _{n\to \infty }\,(a_{n}+b_{n})\geq \liminf _{n\to \infty }a_{n}+\ \liminf _{n\to \infty }b_{n}.} 4473:{\displaystyle \limsup _{n\to \infty }\,(a_{n}+b_{n})\leq \limsup _{n\to \infty }a_{n}+\ \limsup _{n\to \infty }b_{n}.} 1196:

The limits superior and inferior can equivalently be defined using the concept of subsequential limits of the sequence

12331: 12312: 5206: 86: 64: 4224:{\displaystyle \inf _{n}x_{n}\leq \liminf _{n\to \infty }x_{n}\leq \limsup _{n\to \infty }x_{n}\leq \sup _{n}x_{n}.} 2964: 2915: 1756:

of real numbers may not exist (the reals are not a complete lattice), it is convenient to consider sequences in the

57: 10588:) sequence as a whole, and so the interior or inferior limit is the empty set { }. So, as in the previous example, 1805:

consisting of real numbers. Assume that the limit superior and limit inferior are real numbers (so, not infinite).

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Therefore, the limit inferior and limit superior of the net are equal to the limit superior and limit inferior of

11506: 8020: 7746: 4696: 4653: 7352:(i.e., how to quantify separation) is defined. In fact, the second definition is identical to the first when the 5128: 9303:

So the limit infimum contains all subsets which are lower bounds for all but finitely many sets of the sequence.

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The limit superior and limit inferior of a sequence are a special case of those of a function (see below).

7270:. Hence, it is possible (and sometimes useful) to consider superior and inferior limits of sequences in ℘( 3230:

is the smallest closed interval with this property. We can formalize this property like this: there exist

119:(that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a 12354: 11043: 6057: 3671:{\displaystyle \liminf _{n\to \infty }x_{n}-\epsilon <x_{n}<\limsup _{n\to \infty }x_{n}+\epsilon } 7907: 7699: 7554: 2456: 867:{\displaystyle \limsup _{n\to \infty }x_{n}:=\lim _{n\to \infty }\!{\Big (}\sup _{m\geq n}x_{m}{\Big )}} 592:{\displaystyle \liminf _{n\to \infty }x_{n}:=\lim _{n\to \infty }\!{\Big (}\inf _{m\geq n}x_{m}{\Big )}} 11868: 8231:

superior and inferior, as the latter sets are not sensitive to the topological structure of the space.

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of the sequence are maintained, the outer and inner limits will be unchanged. The related concepts of

4763: 4327: 4242: 7316: 7208: 5950: 5353: 5097: 4301: 3122: 1602: 1601:, the limit superior and limit inferior always exist, as the real numbers together with ±∞ (i.e. the 11480: 10305:{\displaystyle (X_{n})=(\{50\},\{20\},\{-100\},\{-25\},\{0\},\{1\},\{0\},\{1\},\{0\},\{1\},\dots ).} 8378:{\displaystyle d(x,y):={\begin{cases}0&{\text{if }}x=y,\\1&{\text{if }}x\neq y,\end{cases}}} 8317: 7980: 7799: 7463: 5792: 5394: 2184: 2177: 2145: 1964: 9854: 8135: 8105: 5977:") are discontinuities which, unless they make up a set of zero, are confined to a negligible set. 5600: 2644: 2303:

The relationship of limit inferior and limit superior for sequences of real numbers is as follows:

465:{\displaystyle \limsup _{n\to \infty }x_{n}\quad {\text{or}}\quad \varlimsup _{n\to \infty }x_{n}.} 343:{\displaystyle \liminf _{n\to \infty }x_{n}\quad {\text{or}}\quad \varliminf _{n\to \infty }x_{n},} 51: 11623: 6461: 2396: 2040: 1859: 11552: 7626: 4617: 3552: 11835: 8075: 5177: 2293:{\displaystyle (\liminf _{n\to \infty }x_{n}-\epsilon ,\limsup _{n\to \infty }x_{n}+\epsilon ).} 10741: 8734: 7369: 7243: 5388: 3807: 3734: 3270: 3236: 120: 68: 5751: 4089: 4069: 4022: 4002: 3952: 3932: 3885: 3865: 3760: 3687: 2669: 11802: 11114: 10924: 10567:{\displaystyle (X_{n})=(\{0\},\{1\},\{1/2\},\{1/2\},\{2/3\},\{1/3\},\{3/4\},\{1/4\},\dots ).} 8738: 7947: 7666: 7590: 7404: 5741: 5596: 2692: 1775: 1745: 1610: 1606: 1448: 1324: 1288: 1250: 1199: 353: 231: 10014:) sequence as a whole, and so the interior or inferior limit is the empty set { }. That is, 3045: 2119: 1938: 11081: 7836: 7500: 5687: 4042: 3975: 3905: 3838: 3780: 3707: 3358: 3331: 3304: 3151: 1993: 1812: 132: 12261:

Goebel, Rafal; Sanfelice, Ricardo G.; Teel, Andrew R. (2009). "Hybrid dynamical systems".

1232: 8: 12057: 10948: 8248: 8228: 7373: 7195: 6506: 5962: 2523: 1761: 221: 124: 116: 7531: 7437: 5330: 3178: 2384:{\displaystyle \limsup _{n\to \infty }\left(-x_{n}\right)=-\liminf _{n\to \infty }x_{n}} 12178: 12131: 12011: 11896: 11782: 11762: 11649: 11486: 11075: 10725: 10355: 8750: 8746: 8477: 7887: 7863: 7525: 7384: 7312: 7301: 7297: 6481: 6455: 6114: 6094: 6037: 6017: 5997: 5745: 5714: 4739: 3075: 2418: 2060: 2020: 1879: 1839: 203:

is shown in blue. The two red curves approach the limit superior and limit inferior of

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There are two common ways to define the limit of sequences of sets. In both cases:

7239: 5958: 1622: 11749:{\displaystyle B:=\{\{x_{\alpha }:\alpha _{0}\leq \alpha \}:\alpha _{0}\in A\}.\,} 1748:, limit superior and limit inferior are important tools for studying sequences of 12052: 12047: 10373: 9846: 8753:

is the least upper bound. In this context, the inner limit, lim inf

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of any net (and thus any sequence) as well. For example, take topological space

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is greater than or equal to the limit inferior; if there are only finitely many

30:"Lower limit" and "upper limit" redirect here. For the statistical concept, see 11216: 10909:{\displaystyle \limsup X:=\sup \,\{x\in Y:x{\text{ is a limit point of }}X\}\,} 10835:{\displaystyle \liminf X:=\inf \,\{x\in Y:x{\text{ is a limit point of }}X\}\,} 10710: 8240: 7311:

The infimum/inferior/inner limit is a set where all of these accumulation sets

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The liminf and limsup of a sequence are respectively the smallest and greatest

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is lesser than or equal to the limit supremum; if there are only finitely many

1729: 1319: 217: 12170: 12123: 11372:{\displaystyle \liminf B:=\inf \,\bigcap \,\{{\overline {B}}_{0}:B_{0}\in B\}} 11193:{\displaystyle \limsup B:=\sup \,\bigcap \,\{{\overline {B}}_{0}:B_{0}\in B\}} 1718:{\displaystyle \liminf _{n\to \infty }x_{n}\leq \limsup _{n\to \infty }x_{n}.} 12370: 10968: 10737: 8749:

of a collection of subsets is the greatest lower bound while the supremum or

5609: 4295: 1753: 1186:{\displaystyle \varlimsup _{n\to \infty }x_{n}:=\limsup _{n\to \infty }x_{n}} 1104:{\displaystyle \varliminf _{n\to \infty }x_{n}:=\liminf _{n\to \infty }x_{n}} 12274: 11386:

is totally ordered, is a complete lattice, and has the order topology, then

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at 0. This idea of oscillation is sufficient to, for example, characterize

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In the particular case that one of the sequences actually converges, say

4232: 3231: 2591:{\displaystyle \liminf _{n\to \infty }x_{n}=\limsup _{n\to \infty }x_{n}} 1749: 1598: 136: 100: 12224:"Lebesgue's Criterion for Riemann integrability (MATH314 Lecture Notes)" 11106: 10934:

in order for these definitions to make sense. Moreover, it has to be a

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The value of this limit inferior is conjectured to be 2 – this is the

11998:{\displaystyle C:=\{\{x_{n}:n_{0}\leq n\}:n_{0}\in \mathbb {N} \}.\,} 10706: 9993:{\displaystyle (X_{n})=(\{0\},\{1\},\{0\},\{1\},\{0\},\{1\},\dots ).} 7259: 7224: 5772: 5748:

to be less than or equal to 246. The corresponding limit superior is

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that are somehow nearby to infinitely many elements of the sequence.

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is greater than the limit superior, there are at most finitely many

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be a topological space. In this case, we replace metric balls with

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whenever the right side of the inequality is defined (that is, not

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An illustration of limit superior and limit inferior. The sequence

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In this sense, the sequence has a limit so long as every point in

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There is a notion of limsup and liminf for functions defined on a

5969:. Note that points of nonzero oscillation (i.e., points at which 10772: 9589:. The greatest lower bound on this sequence of joins of tails is 5320:{\displaystyle \limsup _{n\to \infty }\left(a_{n}b_{n}\right)=AB} 3777:

is less than the limit inferior, there are at most finitely many

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and is similar to the set of limit points of a set. Assume that

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is equal to their common value. (Note that when working just in

5785: 9064:. The least upper bound on this sequence of meets of tails is 11893:. The filter base ("of tails") generated by this sequence is 11030:{\displaystyle \bigcap \,\{{\overline {B}}_{0}:B_{0}\in B\}} 7348:

The difference between the two definitions involves how the

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each of them. Hence, it is the supremum of the limit points.

11479:. Therefore, these definitions give the limit inferior and 11453:{\displaystyle \liminf B=\sup \,\{\inf B_{0}:B_{0}\in B\}.} 11286:{\displaystyle \limsup B=\inf \,\{\sup B_{0}:B_{0}\in B\}.} 8371: 8227:

The outer and inner limits should not be confused with the

7833:, consists of those elements which are limits of points in 7497:, consists of those elements which are limits of points in 7323:

each of them. Hence, it is the infimum of the limit points.

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of the sequence, and the outer limit, lim sup

212:, shown as dashed black lines. In this case, the sequence 5677:{\displaystyle \liminf _{n\to \infty }\,(p_{n+1}-p_{n}),} 11646:. The filter base ("of tails") generated by this net is 11466: 10132:= {50, 20, −100, −25, 0, 1} and the sequence of subsets: 6509:(strictly decreasing or remaining the same), so we have 1739: 8220:{\displaystyle \lim X_{n}=\limsup X_{n}=\liminf X_{n}.} 7190:(there is a way to write the formula using "lim" using 6505:

shrinks, the supremum of the function over the ball is

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Functions from topological spaces to complete lattices

1609:. More generally, these definitions make sense in any 12014: 11922: 11899: 11871: 11838: 11805: 11785: 11765: 11675: 11652: 11626: 11593: 11555: 11509: 11489: 11395: 11309: 11228: 11130: 11084: 11046: 10980: 10862: 10788: 10388: 10144: 9886: 9597: 9333: 9072: 8808: 8290: 8168: 8138: 8108: 8078: 8023: 7983: 7950: 7910: 7890: 7866: 7839: 7802: 7749: 7702: 7669: 7629: 7593: 7557: 7534: 7503: 7466: 7440: 7407: 7387: 7300:

these accumulation sets together. That is, it is the

7015: 6836: 6658: 6517: 6484: 6464: 6425: 6282: 6139: 6117: 6097: 6060: 6040: 6020: 6000: 5901: 5850: 5795: 5754: 5717: 5690: 5618: 5557: 5505: 5453: 5397: 5356: 5333: 5261: 5209: 5180: 5131: 5100: 5094:

hold whenever the right-hand side is not of the form

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almost all sequence members lie in the open interval

2187: 2148: 2122: 2083: 2063: 2043: 2023: 1996: 1967: 1941: 1902: 1882: 1862: 1842: 1815: 1778: 1656: 1545: 1487: 1451: 1438:{\displaystyle E\subseteq {\overline {\mathbb {R} }}} 1416: 1360: 1327: 1291: 1258: 1235: 1202: 1117: 1035: 880: 777: 605: 502: 389: 356: 267: 234: 12302: 8779:of the sequence. The following makes this precise. 6798: 4650:

then the inequalities above become equalities (with

1587:{\displaystyle \liminf _{n\to \infty }x_{n}=\inf E.} 12260: 6803:This finally motivates the definitions for general 4760:For any two sequences of non-negative real numbers 1526:{\displaystyle \limsup _{n\to \infty }x_{n}=\sup E} 12020: 11997: 11905: 11885: 11857: 11824: 11791: 11771: 11748: 11658: 11638: 11612: 11575: 11541: 11495: 11452: 11371: 11285: 11192: 11097: 11066: 11029: 10908: 10834: 10566: 10304: 9992: 9816: 9530: 9291: 9005: 8377: 8219: 8154: 8124: 8094: 8061: 8009: 7969: 7932: 7896: 7872: 7852: 7821: 7787: 7735: 7688: 7655: 7612: 7579: 7543: 7516: 7485: 7449: 7426: 7393: 7198:). This version is often useful in discussions of 7179: 7000: 6788: 6644: 6490: 6470: 6446: 6409: 6266: 6123: 6103: 6080: 6046: 6026: 6006: 5941: 5887: 5836: 5763: 5723: 5703: 5676: 5587: 5543: 5491: 5431: 5387:As an example, consider the sequence given by the 5371: 5342: 5319: 5247: 5195: 5166: 5115: 5083: 4946: 4807: 4748: 4728: 4685: 4642: 4606: 4472: 4345: 4316: 4286: 4223: 4098: 4078: 4058: 4031: 4011: 3991: 3961: 3941: 3921: 3894: 3874: 3854: 3819: 3796: 3769: 3746: 3723: 3696: 3670: 3574: 3541: 3504: 3374: 3347: 3320: 3293: 3259: 3222: 3190: 3167: 3140: 3111: 3064: 3034: 3002: 2953: 2902: 2701: 2681: 2658: 2633: 2590: 2515: 2480: 2445: 2407: 2383: 2292: 2199: 2160: 2134: 2108: 2069: 2049: 2029: 2009: 1979: 1953: 1927: 1888: 1868: 1848: 1828: 1797: 1717: 1586: 1525: 1470: 1437: 1403:{\displaystyle \xi =\lim _{k\to \infty }x_{n_{k}}} 1402: 1346: 1318:if there exists a strictly increasing sequence of 1310: 1273: 1241: 1221: 1185: 1103: 1019: 866: 744: 591: 464: 375: 342: 253: 11296:Similarly, the limit inferior of the filter base 9806: 9281: 7587:if and only if there exists a sequence of points 5044: 5043: 4938: 4906: 4905: 4899: 2393:As mentioned earlier, it is convenient to extend 859: 826: 823: 584: 551: 548: 12368: 11412: 11405: 11396: 11319: 11310: 11245: 11238: 11229: 11140: 11131: 10872: 10863: 10853:of all of the limit points of the set. That is, 10798: 10789: 9643: 9635: 9603: 9355: 9118: 9110: 9078: 8830: 8234: 8201: 8185: 8169: 8139: 8109: 8079: 8025: 7917: 7803: 7751: 7564: 7467: 7296:The supremum/superior/outer limit is a set that 7055: 7047: 7017: 6876: 6868: 6838: 6711: 6691: 6660: 6570: 6550: 6519: 6335: 6315: 6284: 6192: 6172: 6141: 5903: 5852: 5620: 5544:{\displaystyle \limsup _{n\to \infty }x_{n}=+1.} 5507: 5455: 5263: 5217: 5133: 5051: 5013: 4962: 4913: 4874: 4823: 4701: 4658: 4576: 4544: 4495: 4442: 4410: 4361: 4199: 4170: 4141: 4118: 3637: 3589: 3471: 3390: 2975: 2926: 2859: 2808: 2769: 2721: 2606: 2563: 2534: 2356: 2311: 2253: 2218: 1728:The limits inferior and superior are related to 1687: 1658: 1575: 1547: 1517: 1489: 1368: 1158: 1076: 964: 956: 928: 911: 882: 832: 808: 779: 689: 681: 653: 636: 607: 557: 533: 504: 391: 269: 11779:respectively. Similarly, for topological space 5985: 5492:{\displaystyle \liminf _{n\to \infty }x_{n}=-1} 5248:{\displaystyle B=\limsup _{n\to \infty }b_{n},} 3969:is greater than or equal to the limit supremum. 12292:. Princeton, NJ: D. Van Nostrand Company, Inc. 11471:Note that filter bases are generalizations of 10942: 9857:is an example application of these constructs. 4106:is lesser than or equal to the limit inferior. 3003:{\displaystyle S=\limsup _{n\to \infty }x_{n}} 2954:{\displaystyle I=\liminf _{n\to \infty }x_{n}} 11988: 11964: 11932: 11929: 11739: 11717: 11685: 11682: 11542:{\displaystyle (x_{\alpha })_{\alpha \in A}} 11444: 11409: 11366: 11327: 11277: 11242: 11187: 11148: 11024: 10985: 10902: 10876: 10828: 10802: 10549: 10535: 10529: 10515: 10509: 10495: 10489: 10475: 10469: 10455: 10449: 10435: 10429: 10423: 10417: 10411: 10287: 10281: 10275: 10269: 10263: 10257: 10251: 10245: 10239: 10233: 10227: 10221: 10215: 10206: 10200: 10191: 10185: 10179: 10173: 10167: 9975: 9969: 9963: 9957: 9951: 9945: 9939: 9933: 9927: 9921: 9915: 9909: 9729: 9726: 9708: 9696: 9693: 9669: 9647: 9639: 9420: 9417: 9381: 9359: 9204: 9201: 9183: 9171: 9168: 9144: 9122: 9114: 8895: 8892: 8856: 8834: 8062:{\displaystyle \lim _{k\to \infty }x_{k}=x.} 7788:{\displaystyle \lim _{k\to \infty }x_{k}=x.} 7174: 7165: 7159: 7104: 7101: 7095: 7059: 7051: 6995: 6986: 6980: 6925: 6922: 6916: 6880: 6872: 6775: 6772: 6766: 6715: 6634: 6631: 6625: 6574: 6399: 6396: 6390: 6339: 6256: 6253: 6247: 6196: 5582: 5558: 4729:{\displaystyle \liminf _{n\to \infty }a_{n}} 4686:{\displaystyle \limsup _{n\to \infty }a_{n}} 3827:if it is greater, there are infinitely many. 1011: 995: 968: 960: 736: 720: 693: 685: 8733:Using the standard parlance of set theory, 8712:either appears in all except finitely many 8493:if and only if there exists a subsequence ( 5167:{\displaystyle \lim _{n\to \infty }a_{n}=A} 474: 12256: 12254: 12183:: CS1 maint: location missing publisher ( 12136:: CS1 maint: location missing publisher ( 10750: 8659:exists if and only if lim inf 7363: 5744:– but as of April 2014 has only been 4483:Analogously, the limit inferior satisfies 3449: 3448: 3447: 3446: 3445: 3444: 3443: 3442: 3441: 2857: 2856: 2850: 2849: 2767: 2766: 2760: 2759: 1445:is the set of all subsequential limits of 1274:{\displaystyle {\overline {\mathbb {R} }}} 11994: 11984: 11879: 11745: 11408: 11326: 11322: 11241: 11147: 11143: 10984: 10905: 10875: 10831: 10801: 10626:) = ({0}, {1/2}, {2/3}, {3/4}, ...) and ( 9646: 9642: 9638: 9358: 9121: 9117: 9113: 8833: 7146: 7133: 7058: 7054: 7050: 6967: 6954: 6879: 6875: 6871: 6714: 6573: 6338: 6195: 6074: 5784:Assume that a function is defined from a 5635: 4977: 4838: 4510: 4376: 3754:if it is less, there are infinitely many. 3535: 2649: 2634:{\displaystyle \lim _{n\to \infty }x_{n}} 2401: 2181:In case the sequence is bounded, for all 1426: 1262: 1010: 994: 971: 967: 963: 959: 926: 735: 719: 696: 692: 688: 684: 651: 87:Learn how and when to remove this message 12321: 12038:Essential infimum and essential supremum 11117:. The limit superior of the filter base 9862:Using either the discrete metric or the 5942:{\displaystyle \liminf _{x\to 0}f(x)=-1} 5779: 3831:Conversely, it can also be shown that: 2176: 2037:such that, for any positive real number 1856:such that, for any positive real number 189: 50:This article includes a list of general 12281: 12251: 12150: 12103: 10758: 10724:For example, an LTI system that is the 9055:is the intersection of fewer sets than 8721:or appears in all except finitely many 5888:{\displaystyle \limsup _{x\to 0}f(x)=1} 2109:{\displaystyle x_{n}>b-\varepsilon } 1928:{\displaystyle x_{n}<b+\varepsilon } 1752:. Since the supremum and infimum of an 14: 12369: 12287: 10732:systems with an undamped second-order 10354:inner and outer limits, which use the 5771:, because there are arbitrarily large 4239:For any two sequences of real numbers 12083:. New York: McGraw-Hill. p. 56. 12076: 11467:Specialization for sequences and nets 10923:needs to be defined as a subset of a 10635:) = ({1}, {1/2}, {1/3}, {1/4}, ...): 3542:{\displaystyle n_{0}\in \mathbb {N} } 1740:The case of sequences of real numbers 350:and the limit superior of a sequence 224:(i.e., when there is a single limit). 12155:. Boca Raton, FL. pp. 160–182. 12108:. Boca Raton, FL. pp. 176–177. 10372:Consider the sequence of subsets of 9874:= {0,1} and the sequence of subsets: 8741:on the collection of all subsets of 8281:, the discrete metric is defined by 7218: 3042:need not contain any of the numbers 1758:affinely extended real number system 1633:exist. Whenever lim inf 36: 12303:Amann, H.; Escher, Joachim (2005). 12080:Principles of Mathematical Analysis 11067:{\displaystyle {\overline {B}}_{0}} 8443:, then the following always exist: 6081:{\displaystyle f:E\to \mathbb {R} } 24: 9785: 9758: 9613: 9449: 9260: 9233: 9088: 8924: 8388:under which a sequence of points ( 8035: 7933:{\displaystyle x\in \liminf X_{n}} 7761: 7736:{\displaystyle x_{k}\in X_{n_{k}}} 7580:{\displaystyle x\in \limsup X_{n}} 7356:is used to induce the topology on 7171: 7126: 7123: 7120: 7117: 6992: 6947: 6944: 6941: 6938: 5758: 5630: 5517: 5465: 5363: 5273: 5227: 5190: 5143: 5107: 5061: 5023: 4972: 4923: 4884: 4833: 4711: 4668: 4586: 4554: 4505: 4452: 4420: 4371: 4340: 4334: 4311: 4305: 4180: 4151: 3956: 3936: 3889: 3869: 3738: 3691: 3647: 3599: 3514:On the other hand, there exists a 3481: 3400: 3382:are increasing) for which we have 3175:for all but finitely many indices 2985: 2936: 2890: 2869: 2843: 2818: 2797: 2779: 2753: 2731: 2696: 2676: 2616: 2573: 2544: 2507: 2501: 2481:{\displaystyle \left(x_{n}\right)} 2434: 2428: 2366: 2321: 2263: 2228: 1697: 1668: 1557: 1499: 1378: 1168: 1136: 1086: 1054: 892: 818: 789: 617: 543: 514: 441: 401: 319: 279: 163:; limit superior is also known as 56:it lacks sufficient corresponding 25: 12388: 12342: 12153:Fundamentals of abstract analysis 12106:Fundamentals of abstract analysis 11886:{\displaystyle n\in \mathbb {N} } 11203:when that supremum exists. When 10971:for that filter base is given by 10845:Similarly, the limit superior of 10316:As in the previous two examples, 8591:if and only if there exists some 7274:) (i.e., sequences of subsets of 7156: 7092: 6977: 6913: 6799:Functions from topological spaces 6763: 6622: 6447:{\displaystyle B(a,\varepsilon )} 6387: 6244: 5588:{\displaystyle \{1,2,3,\ldots \}} 1767: 1597:If the terms in the sequence are 228:The limit inferior of a sequence 18:Limit superior and limit inferior 11613:{\displaystyle x_{\alpha }\in X} 9580:is the union of fewer sets than 4808:{\displaystyle (a_{n}),(b_{n}),} 4346:{\displaystyle -\infty +\infty } 4287:{\displaystyle (a_{n}),(b_{n}),} 2057:, there exists a natural number 1876:, there exists a natural number 41: 11475:, which are generalizations of 10896: is a limit point of 10822: is a limit point of 8677:agree, in which case lim 8239:This is the definition used in 5773:gaps between consecutive primes 5372:{\displaystyle 0\cdot \infty .} 5116:{\displaystyle 0\cdot \infty .} 4317:{\displaystyle \infty -\infty } 3141:{\displaystyle \epsilon >0,} 1764:, which is a complete lattice. 422: 416: 300: 294: 12216: 12191: 12144: 12097: 12070: 11819: 11806: 11570: 11556: 11524: 11510: 10967:in that space. The set of all 10558: 10408: 10402: 10389: 10296: 10164: 10158: 10145: 10052:) = ({0}, {0}, {0}, ...) and ( 9984: 9906: 9900: 9887: 9610: 9321:tail of the sequence. That is, 9085: 8796:tail of the sequence. That is, 8439:) is a sequence of subsets of 8306: 8294: 8032: 8010:{\displaystyle x_{k}\in X_{k}} 7964: 7951: 7940:if and only if there exists a 7822:{\displaystyle \liminf X_{n},} 7758: 7683: 7670: 7650: 7630: 7607: 7594: 7486:{\displaystyle \limsup X_{n},} 7421: 7408: 7071: 7065: 7041: 7035: 7024: 6892: 6886: 6862: 6856: 6845: 6760: 6748: 6727: 6721: 6684: 6678: 6667: 6619: 6607: 6586: 6580: 6543: 6537: 6526: 6441: 6429: 6384: 6372: 6351: 6345: 6322: 6308: 6302: 6291: 6241: 6229: 6208: 6202: 6179: 6165: 6159: 6148: 6070: 5927: 5921: 5910: 5876: 5870: 5859: 5837:{\displaystyle f(x)=\sin(1/x)} 5831: 5817: 5805: 5799: 5668: 5636: 5627: 5551:(This is because the sequence 5514: 5462: 5432:{\displaystyle x_{n}=\sin(n).} 5423: 5417: 5270: 5224: 5140: 5058: 5020: 5001: 4978: 4969: 4920: 4881: 4862: 4839: 4830: 4799: 4786: 4780: 4767: 4708: 4665: 4631: 4583: 4551: 4537: 4511: 4502: 4449: 4417: 4403: 4377: 4368: 4278: 4265: 4259: 4246: 4177: 4148: 3644: 3596: 3478: 3397: 3217: 3205: 3103: 3079: 3029: 3017: 2982: 2933: 2866: 2815: 2776: 2728: 2613: 2570: 2541: 2510: 2495: 2437: 2422: 2363: 2318: 2284: 2260: 2225: 2214: 2200:{\displaystyle \epsilon >0} 2161:{\displaystyle b-\varepsilon } 1980:{\displaystyle b+\varepsilon } 1792: 1779: 1694: 1665: 1554: 1496: 1465: 1452: 1375: 1341: 1328: 1305: 1292: 1216: 1203: 1165: 1133: 1083: 1051: 889: 815: 786: 614: 540: 511: 438: 398: 370: 357: 316: 276: 248: 235: 13: 1: 12307:. Basel; Boston: Birkhäuser. 12263:IEEE Control Systems Magazine 12199:"Bounded gaps between primes" 12063: 11382:when that infimum exists; if 9832: 8235:Special case: discrete metric 8155:{\displaystyle \limsup X_{n}} 8125:{\displaystyle \liminf X_{n}} 4294:the limit superior satisfies 3972:If there are infinitely many 3835:If there are infinitely many 3072:but every slight enlargement 2659:{\displaystyle \mathbb {R} ,} 2172: 1029:Alternatively, the notations 32:Lower/upper confidence limits 11639:{\displaystyle \alpha \in A} 11336: 11157: 11053: 10994: 10763:The limit inferior of a set 7434:is a sequence of subsets of 6471:{\displaystyle \varepsilon } 5986:Functions from metric spaces 2408:{\displaystyle \mathbb {R} } 2050:{\displaystyle \varepsilon } 1869:{\displaystyle \varepsilon } 1836:is the smallest real number 1430: 1266: 1124: 429: 7: 12355:Encyclopedia of Mathematics 12151:Gleason, Andrew M. (1992). 12104:Gleason, Andrew M. (1992). 12031: 11576:{\displaystyle (A,{\leq })} 10943:Definition for filter bases 7656:{\displaystyle (X_{n_{k}})} 5597:equidistributed mod 2π 5381: 5174:exists (including the case 4643:{\displaystyle a_{n}\to a,} 3575:{\displaystyle n\geq n_{0}} 2017:is the largest real number 10: 12393: 12324:Classical complex analysis 12322:González, Mario O (1991). 11858:{\displaystyle x_{n}\in X} 10946: 10367:Using the Euclidean metric 10061:) = ({1}, {1}, {1}, ...): 9551:) is non-increasing (i.e. 9026:) is non-decreasing (i.e. 8418:for all but finitely many 8095:{\displaystyle \lim X_{n}} 7860:for all but finitely many 7367: 5196:{\displaystyle A=+\infty } 29: 8777:smallest joining of tails 8269:Specifically, for points 7829:which is also called the 7493:which is also called the 7209:extended real number line 3862:greater than or equal to 3820:{\displaystyle \lambda ;} 3747:{\displaystyle \Lambda ;} 3294:{\displaystyle x_{h_{n}}} 3260:{\displaystyle x_{k_{n}}} 1603:extended real number line 12350:"Upper and lower limits" 12288:Halmos, Paul R. (1950). 8764:largest meeting of tails 8668:and lim sup 8565:all except finitely many 8550:consists of elements of 8456:consists of elements of 7378:A sequence of sets in a 5764:{\displaystyle +\infty } 5601:equidistribution theorem 4099:{\displaystyle \lambda } 4079:{\displaystyle \lambda } 4032:{\displaystyle \lambda } 4012:{\displaystyle \lambda } 3999:lesser than or equal to 3962:{\displaystyle \Lambda } 3942:{\displaystyle \Lambda } 3895:{\displaystyle \Lambda } 3875:{\displaystyle \Lambda } 3770:{\displaystyle \lambda } 3697:{\displaystyle \Lambda } 2682:{\displaystyle -\infty } 1640:and lim sup 475:Definition for sequences 12326:. New York: M. Dekker. 12275:10.1109/MCS.2008.931718 11825:{\displaystyle (x_{n})} 10751:Generalized definitions 10579:can be taken from the ( 8424:if the limit set exists 7970:{\displaystyle (x_{k})} 7689:{\displaystyle (X_{n})} 7613:{\displaystyle (x_{k})} 7427:{\displaystyle (X_{n})} 7364:General set convergence 6819:as before, but now let 5599:, a consequence of the 2702:{\displaystyle \infty } 1798:{\displaystyle (x_{n})} 1471:{\displaystyle (x_{n})} 1347:{\displaystyle (n_{k})} 1311:{\displaystyle (x_{n})} 1222:{\displaystyle (x_{n})} 376:{\displaystyle (x_{n})} 254:{\displaystyle (x_{n})} 71:more precise citations. 12022: 11999: 11907: 11887: 11859: 11826: 11793: 11773: 11750: 11660: 11640: 11614: 11577: 11543: 11497: 11454: 11373: 11287: 11194: 11099: 11068: 11031: 10910: 10836: 10677:= lim inf 10647:= lim inf 10568: 10306: 10103:= lim inf 10073:= lim inf 9994: 9818: 9789: 9762: 9532: 9453: 9293: 9264: 9237: 9007: 8928: 8695:= lim inf 8686:= lim sup 8637:∉ lim inf 8624:∈ lim sup 8582:∈ lim inf 8484:∈ lim sup 8379: 8221: 8156: 8126: 8102:exists if and only if 8096: 8063: 8011: 7971: 7934: 7898: 7874: 7854: 7823: 7789: 7737: 7690: 7657: 7614: 7581: 7545: 7526:(countably) infinitely 7518: 7487: 7451: 7428: 7395: 7370:Kuratowski convergence 7335:⊆ lim sup 7181: 7002: 6790: 6646: 6492: 6472: 6448: 6411: 6268: 6125: 6105: 6082: 6048: 6028: 6008: 5943: 5889: 5838: 5765: 5725: 5705: 5678: 5589: 5545: 5493: 5433: 5373: 5344: 5321: 5249: 5197: 5168: 5117: 5085: 4948: 4809: 4750: 4730: 4687: 4644: 4608: 4474: 4347: 4318: 4288: 4225: 4100: 4080: 4060: 4033: 4013: 3993: 3963: 3943: 3923: 3896: 3876: 3856: 3821: 3798: 3771: 3748: 3725: 3698: 3672: 3576: 3543: 3506: 3376: 3349: 3322: 3295: 3261: 3224: 3198:In fact, the interval 3192: 3169: 3142: 3119:for arbitrarily small 3113: 3066: 3065:{\displaystyle x_{n},} 3036: 3004: 2955: 2904: 2703: 2683: 2660: 2635: 2592: 2517: 2482: 2447: 2409: 2385: 2300: 2294: 2201: 2162: 2136: 2135:{\displaystyle n>N} 2110: 2071: 2051: 2031: 2011: 1990:The limit inferior of 1981: 1955: 1954:{\displaystyle n>N} 1929: 1890: 1870: 1850: 1830: 1809:The limit superior of 1799: 1719: 1588: 1527: 1472: 1439: 1404: 1348: 1312: 1275: 1243: 1223: 1187: 1105: 1021: 868: 746: 593: 466: 377: 344: 255: 225: 12231:University of Windsor 12023: 12000: 11908: 11888: 11860: 11827: 11794: 11774: 11751: 11661: 11641: 11615: 11578: 11544: 11498: 11455: 11374: 11288: 11195: 11115:partially ordered set 11100: 11098:{\displaystyle B_{0}} 11069: 11032: 10925:partially ordered set 10911: 10837: 10779:of the set. That is, 10709:) of a solution to a 10569: 10307: 9995: 9819: 9769: 9742: 9533: 9433: 9294: 9244: 9217: 9008: 8908: 8397:) converges to point 8380: 8222: 8162:agree, in which case 8157: 8127: 8097: 8064: 8012: 7972: 7935: 7899: 7875: 7855: 7853:{\displaystyle X_{n}} 7824: 7790: 7738: 7691: 7658: 7615: 7582: 7546: 7519: 7517:{\displaystyle X_{n}} 7488: 7452: 7429: 7396: 7315:. That is, it is the 7182: 7003: 6791: 6647: 6493: 6473: 6449: 6412: 6269: 6126: 6106: 6083: 6049: 6029: 6009: 5944: 5890: 5839: 5780:Real-valued functions 5766: 5742:twin prime conjecture 5726: 5706: 5704:{\displaystyle p_{n}} 5679: 5590: 5546: 5494: 5434: 5374: 5345: 5322: 5250: 5198: 5169: 5118: 5086: 4949: 4810: 4751: 4731: 4688: 4645: 4609: 4475: 4348: 4319: 4289: 4226: 4101: 4081: 4061: 4059:{\displaystyle x_{n}} 4034: 4014: 3994: 3992:{\displaystyle x_{n}} 3964: 3944: 3924: 3922:{\displaystyle x_{n}} 3897: 3877: 3857: 3855:{\displaystyle x_{n}} 3822: 3799: 3797:{\displaystyle x_{n}} 3772: 3749: 3726: 3724:{\displaystyle x_{n}} 3699: 3673: 3577: 3544: 3507: 3377: 3375:{\displaystyle h_{n}} 3350: 3348:{\displaystyle k_{n}} 3323: 3321:{\displaystyle x_{n}} 3296: 3262: 3225: 3193: 3170: 3168:{\displaystyle x_{n}} 3143: 3114: 3067: 3037: 3005: 2956: 2905: 2704: 2684: 2661: 2636: 2593: 2518: 2483: 2448: 2410: 2386: 2295: 2202: 2180: 2163: 2137: 2111: 2072: 2052: 2032: 2012: 2010:{\displaystyle x_{n}} 1982: 1956: 1930: 1891: 1871: 1851: 1831: 1829:{\displaystyle x_{n}} 1800: 1746:mathematical analysis 1720: 1611:partially ordered set 1589: 1528: 1473: 1440: 1405: 1349: 1313: 1276: 1251:extended real numbers 1244: 1224: 1188: 1106: 1022: 869: 747: 594: 467: 378: 345: 256: 193: 115:can be thought of as 12377:Limits (mathematics) 12012: 11920: 11897: 11869: 11836: 11803: 11799:, take the sequence 11783: 11763: 11673: 11650: 11624: 11591: 11553: 11507: 11487: 11393: 11307: 11226: 11128: 11105:. This is clearly a 11082: 11044: 10978: 10860: 10786: 10759:Definition for a set 10668:lim sup 10638:lim sup 10603:lim inf 10591:lim sup 10386: 10331:lim inf 10319:lim sup 10142: 10094:lim sup 10064:lim sup 10029:lim inf 10017:lim sup 9884: 9855:Borel–Cantelli lemma 9595: 9331: 9070: 8806: 8541:lim inf 8447:lim sup 8288: 8262:is induced from the 8229:set-theoretic limits 8166: 8136: 8106: 8076: 8021: 7981: 7948: 7908: 7888: 7864: 7837: 7800: 7747: 7700: 7667: 7627: 7591: 7555: 7532: 7501: 7464: 7438: 7405: 7385: 7254:is bounded above by 7013: 6834: 6656: 6515: 6482: 6462: 6423: 6280: 6137: 6115: 6095: 6058: 6038: 6018: 5998: 5899: 5848: 5793: 5752: 5715: 5688: 5616: 5555: 5503: 5451: 5439:Using the fact that 5395: 5354: 5331: 5259: 5207: 5178: 5129: 5098: 4958: 4819: 4764: 4740: 4697: 4654: 4618: 4491: 4357: 4328: 4302: 4243: 4114: 4090: 4070: 4043: 4023: 4003: 3976: 3953: 3933: 3906: 3886: 3866: 3839: 3808: 3781: 3761: 3735: 3708: 3688: 3585: 3553: 3518: 3386: 3359: 3332: 3305: 3271: 3237: 3202: 3179: 3152: 3123: 3076: 3046: 3014: 3010:, then the interval 2965: 2916: 2713: 2693: 2670: 2645: 2602: 2530: 2492: 2457: 2419: 2397: 2307: 2211: 2185: 2146: 2120: 2081: 2061: 2041: 2021: 1994: 1965: 1939: 1900: 1880: 1860: 1840: 1813: 1776: 1772:Consider a sequence 1654: 1647:both exist, we have 1621:exist, such as in a 1543: 1485: 1449: 1414: 1358: 1325: 1289: 1256: 1242:{\displaystyle \xi } 1233: 1200: 1193:are sometimes used. 1115: 1033: 878: 775: 603: 500: 387: 354: 265: 232: 133:infimum and supremum 27:Bounds of a sequence 12058:Set-theoretic limit 10949:Filters in topology 10705:The Ω limit (i.e., 9317:be the join of the 8792:be the meet of the 8249:set-theoretic limit 7374:Subsequential limit 7242:that is ordered by 7215: ∪ {∞}.) 7196:neighborhood filter 5965:except on a set of 5350:is not of the form 2853: implies 2763: implies 1762:totally ordered set 1283:subsequential limit 125:limit of a function 12077:Rudin, W. (1976). 12018: 11995: 11903: 11883: 11855: 11822: 11789: 11769: 11746: 11656: 11636: 11610: 11573: 11539: 11493: 11450: 11369: 11283: 11190: 11095: 11064: 11027: 10919:Note that the set 10906: 10832: 10726:cascade connection 10564: 10356:essential supremum 10302: 9990: 9814: 9812: 9617: 9528: 9526: 9289: 9287: 9092: 9003: 9001: 8478:countably infinite 8375: 8370: 8217: 8152: 8122: 8092: 8059: 8039: 8007: 7967: 7930: 7894: 7870: 7850: 7819: 7785: 7765: 7733: 7686: 7653: 7610: 7577: 7544:{\displaystyle n.} 7541: 7514: 7483: 7450:{\displaystyle X,} 7447: 7424: 7391: 7177: 7031: 6998: 6852: 6805:topological spaces 6786: 6705: 6674: 6642: 6564: 6533: 6488: 6468: 6444: 6407: 6329: 6298: 6264: 6186: 6155: 6121: 6101: 6088:. Define, for any 6078: 6044: 6024: 6004: 5959:Riemann-integrable 5939: 5917: 5885: 5866: 5834: 5761: 5721: 5701: 5674: 5634: 5585: 5541: 5521: 5489: 5469: 5447:, it follows that 5429: 5369: 5343:{\displaystyle AB} 5340: 5317: 5277: 5245: 5231: 5193: 5164: 5147: 5113: 5081: 5065: 5027: 4976: 4944: 4927: 4888: 4837: 4805: 4746: 4736:being replaced by 4726: 4715: 4683: 4672: 4640: 4604: 4590: 4558: 4509: 4470: 4456: 4424: 4375: 4343: 4314: 4284: 4221: 4207: 4184: 4155: 4126: 4096: 4076: 4056: 4029: 4009: 3989: 3959: 3939: 3919: 3892: 3872: 3852: 3817: 3794: 3767: 3744: 3721: 3694: 3668: 3651: 3603: 3572: 3539: 3502: 3485: 3404: 3372: 3345: 3318: 3291: 3257: 3220: 3191:{\displaystyle n.} 3188: 3165: 3138: 3109: 3062: 3032: 3000: 2989: 2951: 2940: 2900: 2898: 2873: 2822: 2783: 2735: 2699: 2679: 2656: 2631: 2620: 2588: 2577: 2548: 2513: 2478: 2443: 2405: 2381: 2370: 2325: 2301: 2290: 2267: 2232: 2197: 2158: 2132: 2106: 2067: 2047: 2027: 2007: 1977: 1951: 1925: 1886: 1866: 1846: 1826: 1795: 1715: 1701: 1672: 1584: 1561: 1523: 1503: 1468: 1435: 1400: 1382: 1344: 1308: 1271: 1239: 1219: 1183: 1172: 1140: 1101: 1090: 1058: 1045: 1017: 942: 925: 896: 864: 846: 822: 793: 742: 667: 650: 621: 589: 571: 547: 518: 462: 445: 405: 373: 340: 323: 310: 283: 251: 226: 12162:978-1-4398-6481-4 12115:978-1-4398-6481-4 12021:{\displaystyle C} 11906:{\displaystyle C} 11792:{\displaystyle X} 11772:{\displaystyle B} 11659:{\displaystyle B} 11496:{\displaystyle X} 11339: 11160: 11056: 10997: 10955:topological space 10932:topological space 10897: 10823: 10717:to the limit set. 10360:essential infimum 10128:Consider the set 9870:Consider the set 9602: 9077: 8595:> 0 such that 8354: 8328: 8024: 7897:{\displaystyle n} 7873:{\displaystyle n} 7750: 7394:{\displaystyle X} 7258:and below by the 7219:Sequences of sets 7115: 7016: 6936: 6837: 6690: 6659: 6549: 6518: 6491:{\displaystyle a} 6314: 6283: 6171: 6140: 6124:{\displaystyle E} 6104:{\displaystyle a} 6054:, and a function 6047:{\displaystyle X} 6027:{\displaystyle E} 6007:{\displaystyle X} 5902: 5851: 5724:{\displaystyle n} 5619: 5506: 5454: 5262: 5216: 5132: 5050: 5012: 4961: 4912: 4873: 4822: 4815:the inequalities 4749:{\displaystyle a} 4700: 4657: 4575: 4574: 4543: 4494: 4441: 4440: 4409: 4360: 4198: 4169: 4140: 4117: 3680:To recapitulate: 3636: 3588: 3470: 3389: 3112:{\displaystyle ,} 2974: 2925: 2858: 2854: 2807: 2768: 2764: 2720: 2605: 2562: 2533: 2446:{\displaystyle .} 2355: 2310: 2252: 2217: 2070:{\displaystyle N} 2030:{\displaystyle b} 1889:{\displaystyle N} 1849:{\displaystyle b} 1686: 1657: 1546: 1488: 1433: 1367: 1269: 1157: 1127: 1118: 1075: 1038: 1036: 927: 910: 881: 831: 807: 778: 652: 635: 606: 556: 532: 503: 432: 423: 420: 390: 303: 301: 298: 268: 97: 96: 89: 16:(Redirected from 12384: 12363: 12337: 12318: 12294: 12293: 12285: 12279: 12278: 12258: 12249: 12248: 12246: 12245: 12239: 12233:. Archived from 12228: 12220: 12214: 12213: 12211: 12209: 12195: 12189: 12188: 12182: 12174: 12148: 12142: 12141: 12135: 12127: 12101: 12095: 12094: 12074: 12053:Dini derivatives 12043:Envelope (waves) 12027: 12025: 12024: 12019: 12004: 12002: 12001: 11996: 11987: 11979: 11978: 11957: 11956: 11944: 11943: 11912: 11910: 11909: 11904: 11892: 11890: 11889: 11884: 11882: 11864: 11862: 11861: 11856: 11848: 11847: 11831: 11829: 11828: 11823: 11818: 11817: 11798: 11796: 11795: 11790: 11778: 11776: 11775: 11770: 11755: 11753: 11752: 11747: 11732: 11731: 11710: 11709: 11697: 11696: 11665: 11663: 11662: 11657: 11645: 11643: 11642: 11637: 11619: 11617: 11616: 11611: 11603: 11602: 11582: 11580: 11579: 11574: 11569: 11548: 11546: 11545: 11540: 11538: 11537: 11522: 11521: 11502: 11500: 11499: 11494: 11459: 11457: 11456: 11451: 11437: 11436: 11424: 11423: 11378: 11376: 11375: 11370: 11359: 11358: 11346: 11345: 11340: 11332: 11292: 11290: 11289: 11284: 11270: 11269: 11257: 11256: 11213:complete lattice 11199: 11197: 11196: 11191: 11180: 11179: 11167: 11166: 11161: 11153: 11104: 11102: 11101: 11096: 11094: 11093: 11073: 11071: 11070: 11065: 11063: 11062: 11057: 11049: 11036: 11034: 11033: 11028: 11017: 11016: 11004: 11003: 10998: 10990: 10936:complete lattice 10915: 10913: 10912: 10907: 10898: 10895: 10841: 10839: 10838: 10833: 10824: 10821: 10573: 10571: 10570: 10565: 10545: 10525: 10505: 10485: 10465: 10445: 10401: 10400: 10374:rational numbers 10311: 10309: 10308: 10303: 10157: 10156: 9999: 9997: 9996: 9991: 9899: 9898: 9864:Euclidean metric 9823: 9821: 9820: 9815: 9813: 9805: 9801: 9800: 9799: 9790: 9788: 9783: 9761: 9756: 9735: 9659: 9658: 9627: 9626: 9616: 9537: 9535: 9534: 9529: 9527: 9514: 9513: 9495: 9494: 9476: 9475: 9463: 9462: 9452: 9447: 9426: 9371: 9370: 9347: 9346: 9298: 9296: 9295: 9290: 9288: 9280: 9276: 9275: 9274: 9265: 9263: 9258: 9236: 9231: 9210: 9134: 9133: 9102: 9101: 9091: 9012: 9010: 9009: 9004: 9002: 8989: 8988: 8970: 8969: 8951: 8950: 8938: 8937: 8927: 8922: 8901: 8846: 8845: 8822: 8821: 8739:partial ordering 8554:which belong to 8460:which belong to 8384: 8382: 8381: 8376: 8374: 8373: 8355: 8352: 8329: 8326: 8226: 8224: 8223: 8218: 8213: 8212: 8197: 8196: 8181: 8180: 8161: 8159: 8158: 8153: 8151: 8150: 8131: 8129: 8128: 8123: 8121: 8120: 8101: 8099: 8098: 8093: 8091: 8090: 8068: 8066: 8065: 8060: 8049: 8048: 8038: 8016: 8014: 8013: 8008: 8006: 8005: 7993: 7992: 7976: 7974: 7973: 7968: 7963: 7962: 7939: 7937: 7936: 7931: 7929: 7928: 7903: 7901: 7900: 7895: 7879: 7877: 7876: 7871: 7859: 7857: 7856: 7851: 7849: 7848: 7828: 7826: 7825: 7820: 7815: 7814: 7794: 7792: 7791: 7786: 7775: 7774: 7764: 7742: 7740: 7739: 7734: 7732: 7731: 7730: 7729: 7712: 7711: 7695: 7693: 7692: 7687: 7682: 7681: 7662: 7660: 7659: 7654: 7649: 7648: 7647: 7646: 7619: 7617: 7616: 7611: 7606: 7605: 7586: 7584: 7583: 7578: 7576: 7575: 7550: 7548: 7547: 7542: 7523: 7521: 7520: 7515: 7513: 7512: 7492: 7490: 7489: 7484: 7479: 7478: 7456: 7454: 7453: 7448: 7433: 7431: 7430: 7425: 7420: 7419: 7400: 7398: 7397: 7392: 7380:metrizable space 7240:complete lattice 7186: 7184: 7183: 7178: 7129: 7113: 7030: 7007: 7005: 7004: 6999: 6950: 6934: 6851: 6795: 6793: 6792: 6787: 6782: 6778: 6704: 6673: 6651: 6649: 6648: 6643: 6641: 6637: 6563: 6532: 6497: 6495: 6494: 6489: 6477: 6475: 6474: 6469: 6453: 6451: 6450: 6445: 6416: 6414: 6413: 6408: 6406: 6402: 6328: 6297: 6273: 6271: 6270: 6265: 6263: 6259: 6185: 6154: 6130: 6128: 6127: 6122: 6110: 6108: 6107: 6102: 6087: 6085: 6084: 6079: 6077: 6053: 6051: 6050: 6045: 6033: 6031: 6030: 6025: 6013: 6011: 6010: 6005: 5948: 5946: 5945: 5940: 5916: 5894: 5892: 5891: 5886: 5865: 5843: 5841: 5840: 5835: 5827: 5770: 5768: 5767: 5762: 5730: 5728: 5727: 5722: 5710: 5708: 5707: 5702: 5700: 5699: 5683: 5681: 5680: 5675: 5667: 5666: 5654: 5653: 5633: 5608:An example from 5594: 5592: 5591: 5586: 5550: 5548: 5547: 5542: 5531: 5530: 5520: 5498: 5496: 5495: 5490: 5479: 5478: 5468: 5438: 5436: 5435: 5430: 5407: 5406: 5378: 5376: 5375: 5370: 5349: 5347: 5346: 5341: 5326: 5324: 5323: 5318: 5307: 5303: 5302: 5301: 5292: 5291: 5276: 5254: 5252: 5251: 5246: 5241: 5240: 5230: 5202: 5200: 5199: 5194: 5173: 5171: 5170: 5165: 5157: 5156: 5146: 5122: 5120: 5119: 5114: 5090: 5088: 5087: 5082: 5080: 5076: 5075: 5074: 5064: 5042: 5038: 5037: 5036: 5026: 5000: 4999: 4990: 4989: 4975: 4953: 4951: 4950: 4945: 4943: 4939: 4937: 4936: 4926: 4904: 4900: 4898: 4897: 4887: 4861: 4860: 4851: 4850: 4836: 4814: 4812: 4811: 4806: 4798: 4797: 4779: 4778: 4755: 4753: 4752: 4747: 4735: 4733: 4732: 4727: 4725: 4724: 4714: 4692: 4690: 4689: 4684: 4682: 4681: 4671: 4649: 4647: 4646: 4641: 4630: 4629: 4613: 4611: 4610: 4605: 4600: 4599: 4589: 4572: 4568: 4567: 4557: 4536: 4535: 4523: 4522: 4508: 4479: 4477: 4476: 4471: 4466: 4465: 4455: 4438: 4434: 4433: 4423: 4402: 4401: 4389: 4388: 4374: 4352: 4350: 4349: 4344: 4323: 4321: 4320: 4315: 4293: 4291: 4290: 4285: 4277: 4276: 4258: 4257: 4230: 4228: 4227: 4222: 4217: 4216: 4206: 4194: 4193: 4183: 4165: 4164: 4154: 4136: 4135: 4125: 4105: 4103: 4102: 4097: 4085: 4083: 4082: 4077: 4065: 4063: 4062: 4057: 4055: 4054: 4038: 4036: 4035: 4030: 4018: 4016: 4015: 4010: 3998: 3996: 3995: 3990: 3988: 3987: 3968: 3966: 3965: 3960: 3948: 3946: 3945: 3940: 3928: 3926: 3925: 3920: 3918: 3917: 3901: 3899: 3898: 3893: 3881: 3879: 3878: 3873: 3861: 3859: 3858: 3853: 3851: 3850: 3826: 3824: 3823: 3818: 3803: 3801: 3800: 3795: 3793: 3792: 3776: 3774: 3773: 3768: 3753: 3751: 3750: 3745: 3730: 3728: 3727: 3722: 3720: 3719: 3703: 3701: 3700: 3695: 3677: 3675: 3674: 3669: 3661: 3660: 3650: 3632: 3631: 3613: 3612: 3602: 3581: 3579: 3578: 3573: 3571: 3570: 3549:so that for all 3548: 3546: 3545: 3540: 3538: 3530: 3529: 3511: 3509: 3508: 3503: 3495: 3494: 3484: 3466: 3465: 3464: 3463: 3440: 3439: 3438: 3437: 3414: 3413: 3403: 3381: 3379: 3378: 3373: 3371: 3370: 3354: 3352: 3351: 3346: 3344: 3343: 3327: 3325: 3324: 3319: 3317: 3316: 3300: 3298: 3297: 3292: 3290: 3289: 3288: 3287: 3266: 3264: 3263: 3258: 3256: 3255: 3254: 3253: 3229: 3227: 3226: 3223:{\displaystyle } 3221: 3197: 3195: 3194: 3189: 3174: 3172: 3171: 3166: 3164: 3163: 3147: 3145: 3144: 3139: 3118: 3116: 3115: 3110: 3071: 3069: 3068: 3063: 3058: 3057: 3041: 3039: 3038: 3035:{\displaystyle } 3033: 3009: 3007: 3006: 3001: 2999: 2998: 2988: 2960: 2958: 2957: 2952: 2950: 2949: 2939: 2909: 2907: 2906: 2901: 2899: 2883: 2882: 2872: 2855: 2852: 2847: 2832: 2831: 2821: 2793: 2792: 2782: 2765: 2762: 2757: 2745: 2744: 2734: 2708: 2706: 2705: 2700: 2688: 2686: 2685: 2680: 2665: 2663: 2662: 2657: 2652: 2640: 2638: 2637: 2632: 2630: 2629: 2619: 2597: 2595: 2594: 2589: 2587: 2586: 2576: 2558: 2557: 2547: 2522: 2520: 2519: 2516:{\displaystyle } 2514: 2487: 2485: 2484: 2479: 2477: 2473: 2472: 2452: 2450: 2449: 2444: 2414: 2412: 2411: 2406: 2404: 2390: 2388: 2387: 2382: 2380: 2379: 2369: 2348: 2344: 2343: 2342: 2324: 2299: 2297: 2296: 2291: 2277: 2276: 2266: 2242: 2241: 2231: 2206: 2204: 2203: 2198: 2167: 2165: 2164: 2159: 2141: 2139: 2138: 2133: 2115: 2113: 2112: 2107: 2093: 2092: 2076: 2074: 2073: 2068: 2056: 2054: 2053: 2048: 2036: 2034: 2033: 2028: 2016: 2014: 2013: 2008: 2006: 2005: 1986: 1984: 1983: 1978: 1960: 1958: 1957: 1952: 1934: 1932: 1931: 1926: 1912: 1911: 1895: 1893: 1892: 1887: 1875: 1873: 1872: 1867: 1855: 1853: 1852: 1847: 1835: 1833: 1832: 1827: 1825: 1824: 1804: 1802: 1801: 1796: 1791: 1790: 1724: 1722: 1721: 1716: 1711: 1710: 1700: 1682: 1681: 1671: 1623:complete lattice 1593: 1591: 1590: 1585: 1571: 1570: 1560: 1532: 1530: 1529: 1524: 1513: 1512: 1502: 1477: 1475: 1474: 1469: 1464: 1463: 1444: 1442: 1441: 1436: 1434: 1429: 1424: 1409: 1407: 1406: 1401: 1399: 1398: 1397: 1396: 1381: 1353: 1351: 1350: 1345: 1340: 1339: 1317: 1315: 1314: 1309: 1304: 1303: 1280: 1278: 1277: 1272: 1270: 1265: 1260: 1248: 1246: 1245: 1240: 1228: 1226: 1225: 1220: 1215: 1214: 1192: 1190: 1189: 1184: 1182: 1181: 1171: 1153: 1152: 1139: 1128: 1120: 1110: 1108: 1107: 1102: 1100: 1099: 1089: 1071: 1070: 1057: 1046: 1026: 1024: 1023: 1018: 981: 980: 952: 951: 941: 924: 906: 905: 895: 873: 871: 870: 865: 863: 862: 856: 855: 845: 830: 829: 821: 803: 802: 792: 771:) is defined by 760: 759: 751: 749: 748: 743: 706: 705: 677: 676: 666: 649: 631: 630: 620: 598: 596: 595: 590: 588: 587: 581: 580: 570: 555: 554: 546: 528: 527: 517: 496:) is defined by 485: 484: 471: 469: 468: 463: 458: 457: 444: 433: 425: 421: 418: 415: 414: 404: 382: 380: 379: 374: 369: 368: 349: 347: 346: 341: 336: 335: 322: 311: 299: 296: 293: 292: 282: 260: 258: 257: 252: 247: 246: 220:the sequence is 92: 85: 81: 78: 72: 67:this article by 58:inline citations 45: 44: 37: 21: 12392: 12391: 12387: 12386: 12385: 12383: 12382: 12381: 12367: 12366: 12348: 12345: 12340: 12334: 12315: 12298: 12297: 12286: 12282: 12259: 12252: 12243: 12241: 12237: 12226: 12222: 12221: 12217: 12207: 12205: 12197: 12196: 12192: 12176: 12175: 12163: 12149: 12145: 12129: 12128: 12116: 12102: 12098: 12091: 12075: 12071: 12066: 12048:One-sided limit 12034: 12013: 12010: 12009: 11983: 11974: 11970: 11952: 11948: 11939: 11935: 11921: 11918: 11917: 11898: 11895: 11894: 11878: 11870: 11867: 11866: 11843: 11839: 11837: 11834: 11833: 11813: 11809: 11804: 11801: 11800: 11784: 11781: 11780: 11764: 11761: 11760: 11727: 11723: 11705: 11701: 11692: 11688: 11674: 11671: 11670: 11651: 11648: 11647: 11625: 11622: 11621: 11598: 11594: 11592: 11589: 11588: 11565: 11554: 11551: 11550: 11527: 11523: 11517: 11513: 11508: 11505: 11504: 11488: 11485: 11484: 11469: 11432: 11428: 11419: 11415: 11394: 11391: 11390: 11354: 11350: 11341: 11331: 11330: 11308: 11305: 11304: 11265: 11261: 11252: 11248: 11227: 11224: 11223: 11175: 11171: 11162: 11152: 11151: 11129: 11126: 11125: 11089: 11085: 11083: 11080: 11079: 11058: 11048: 11047: 11045: 11042: 11041: 11012: 11008: 10999: 10989: 10988: 10979: 10976: 10975: 10951: 10945: 10930:that is also a 10894: 10861: 10858: 10857: 10820: 10787: 10784: 10783: 10761: 10753: 10694: 10685: 10676: 10664: 10655: 10646: 10634: 10625: 10611: 10599: 10587: 10541: 10521: 10501: 10481: 10461: 10441: 10396: 10392: 10387: 10384: 10383: 10339: 10327: 10152: 10148: 10143: 10140: 10139: 10120: 10111: 10102: 10090: 10081: 10072: 10060: 10051: 10037: 10025: 10013: 9894: 9890: 9885: 9882: 9881: 9847:discrete metric 9835: 9811: 9810: 9795: 9791: 9784: 9773: 9768: 9767: 9763: 9757: 9746: 9733: 9732: 9654: 9650: 9628: 9622: 9618: 9606: 9598: 9596: 9593: 9592: 9588: 9579: 9570:) because each 9569: 9559: 9550: 9525: 9524: 9503: 9499: 9484: 9480: 9471: 9467: 9458: 9454: 9448: 9437: 9424: 9423: 9366: 9362: 9348: 9342: 9338: 9334: 9332: 9329: 9328: 9316: 9308:Similarly, let 9286: 9285: 9270: 9266: 9259: 9248: 9243: 9242: 9238: 9232: 9221: 9208: 9207: 9129: 9125: 9103: 9097: 9093: 9081: 9073: 9071: 9068: 9067: 9063: 9054: 9045:) because each 9044: 9034: 9025: 9000: 8999: 8978: 8974: 8959: 8955: 8946: 8942: 8933: 8929: 8923: 8912: 8899: 8898: 8841: 8837: 8823: 8817: 8813: 8809: 8807: 8804: 8803: 8791: 8774: 8761: 8729: 8720: 8703: 8694: 8685: 8676: 8667: 8658: 8645: 8633:if and only if 8632: 8607: 8590: 8562: 8549: 8533: 8532: 8515: 8506: 8505: 8492: 8471:infinitely many 8468: 8455: 8438: 8413: 8405:if and only if 8396: 8369: 8368: 8351: 8349: 8343: 8342: 8325: 8323: 8313: 8312: 8289: 8286: 8285: 8264:discrete metric 8237: 8208: 8204: 8192: 8188: 8176: 8172: 8167: 8164: 8163: 8146: 8142: 8137: 8134: 8133: 8116: 8112: 8107: 8104: 8103: 8086: 8082: 8077: 8074: 8073: 8044: 8040: 8028: 8022: 8019: 8018: 8001: 7997: 7988: 7984: 7982: 7979: 7978: 7958: 7954: 7949: 7946: 7945: 7924: 7920: 7909: 7906: 7905: 7889: 7886: 7885: 7865: 7862: 7861: 7844: 7840: 7838: 7835: 7834: 7810: 7806: 7801: 7798: 7797: 7770: 7766: 7754: 7748: 7745: 7744: 7725: 7721: 7720: 7716: 7707: 7703: 7701: 7698: 7697: 7677: 7673: 7668: 7665: 7664: 7642: 7638: 7637: 7633: 7628: 7625: 7624: 7601: 7597: 7592: 7589: 7588: 7571: 7567: 7556: 7553: 7552: 7533: 7530: 7529: 7508: 7504: 7502: 7499: 7498: 7474: 7470: 7465: 7462: 7461: 7439: 7436: 7435: 7415: 7411: 7406: 7403: 7402: 7386: 7383: 7382: 7376: 7366: 7354:discrete metric 7343: 7334: 7221: 7200:semi-continuity 7116: 7020: 7014: 7011: 7010: 6937: 6841: 6835: 6832: 6831: 6801: 6710: 6706: 6694: 6663: 6657: 6654: 6653: 6569: 6565: 6553: 6522: 6516: 6513: 6512: 6483: 6480: 6479: 6463: 6460: 6459: 6424: 6421: 6420: 6334: 6330: 6318: 6287: 6281: 6278: 6277: 6191: 6187: 6175: 6144: 6138: 6135: 6134: 6116: 6113: 6112: 6096: 6093: 6092: 6073: 6059: 6056: 6055: 6039: 6036: 6035: 6019: 6016: 6015: 5999: 5996: 5995: 5988: 5983: 5906: 5900: 5897: 5896: 5855: 5849: 5846: 5845: 5823: 5794: 5791: 5790: 5789:example, given 5782: 5753: 5750: 5749: 5716: 5713: 5712: 5695: 5691: 5689: 5686: 5685: 5662: 5658: 5643: 5639: 5623: 5617: 5614: 5613: 5556: 5553: 5552: 5526: 5522: 5510: 5504: 5501: 5500: 5474: 5470: 5458: 5452: 5449: 5448: 5402: 5398: 5396: 5393: 5392: 5384: 5355: 5352: 5351: 5332: 5329: 5328: 5297: 5293: 5287: 5283: 5282: 5278: 5266: 5260: 5257: 5256: 5236: 5232: 5220: 5208: 5205: 5204: 5179: 5176: 5175: 5152: 5148: 5136: 5130: 5127: 5126: 5099: 5096: 5095: 5070: 5066: 5054: 5049: 5045: 5032: 5028: 5016: 5011: 5007: 4995: 4991: 4985: 4981: 4965: 4959: 4956: 4955: 4932: 4928: 4916: 4911: 4907: 4893: 4889: 4877: 4872: 4868: 4856: 4852: 4846: 4842: 4826: 4820: 4817: 4816: 4793: 4789: 4774: 4770: 4765: 4762: 4761: 4741: 4738: 4737: 4720: 4716: 4704: 4698: 4695: 4694: 4677: 4673: 4661: 4655: 4652: 4651: 4625: 4621: 4619: 4616: 4615: 4595: 4591: 4579: 4563: 4559: 4547: 4531: 4527: 4518: 4514: 4498: 4492: 4489: 4488: 4485:superadditivity 4461: 4457: 4445: 4429: 4425: 4413: 4397: 4393: 4384: 4380: 4364: 4358: 4355: 4354: 4329: 4326: 4325: 4303: 4300: 4299: 4272: 4268: 4253: 4249: 4244: 4241: 4240: 4212: 4208: 4202: 4189: 4185: 4173: 4160: 4156: 4144: 4131: 4127: 4121: 4115: 4112: 4111: 4091: 4088: 4087: 4071: 4068: 4067: 4050: 4046: 4044: 4041: 4040: 4024: 4021: 4020: 4004: 4001: 4000: 3983: 3979: 3977: 3974: 3973: 3954: 3951: 3950: 3934: 3931: 3930: 3913: 3909: 3907: 3904: 3903: 3887: 3884: 3883: 3867: 3864: 3863: 3846: 3842: 3840: 3837: 3836: 3809: 3806: 3805: 3788: 3784: 3782: 3779: 3778: 3762: 3759: 3758: 3736: 3733: 3732: 3715: 3711: 3709: 3706: 3705: 3689: 3686: 3685: 3656: 3652: 3640: 3627: 3623: 3608: 3604: 3592: 3586: 3583: 3582: 3566: 3562: 3554: 3551: 3550: 3534: 3525: 3521: 3519: 3516: 3515: 3490: 3486: 3474: 3459: 3455: 3454: 3450: 3433: 3429: 3428: 3424: 3409: 3405: 3393: 3387: 3384: 3383: 3366: 3362: 3360: 3357: 3356: 3339: 3335: 3333: 3330: 3329: 3312: 3308: 3306: 3303: 3302: 3283: 3279: 3278: 3274: 3272: 3269: 3268: 3249: 3245: 3244: 3240: 3238: 3235: 3234: 3203: 3200: 3199: 3180: 3177: 3176: 3159: 3155: 3153: 3150: 3149: 3124: 3121: 3120: 3077: 3074: 3073: 3053: 3049: 3047: 3044: 3043: 3015: 3012: 3011: 2994: 2990: 2978: 2966: 2963: 2962: 2945: 2941: 2929: 2917: 2914: 2913: 2897: 2896: 2878: 2874: 2862: 2851: 2846: 2833: 2827: 2823: 2811: 2804: 2803: 2788: 2784: 2772: 2761: 2756: 2746: 2740: 2736: 2724: 2716: 2714: 2711: 2710: 2694: 2691: 2690: 2671: 2668: 2667: 2666:convergence to 2648: 2646: 2643: 2642: 2625: 2621: 2609: 2603: 2600: 2599: 2582: 2578: 2566: 2553: 2549: 2537: 2531: 2528: 2527: 2526:if and only if 2493: 2490: 2489: 2468: 2464: 2460: 2458: 2455: 2454: 2420: 2417: 2416: 2400: 2398: 2395: 2394: 2375: 2371: 2359: 2338: 2334: 2330: 2326: 2314: 2308: 2305: 2304: 2272: 2268: 2256: 2237: 2233: 2221: 2212: 2209: 2208: 2186: 2183: 2182: 2175: 2147: 2144: 2143: 2121: 2118: 2117: 2088: 2084: 2082: 2079: 2078: 2062: 2059: 2058: 2042: 2039: 2038: 2022: 2019: 2018: 2001: 1997: 1995: 1992: 1991: 1966: 1963: 1962: 1940: 1937: 1936: 1907: 1903: 1901: 1898: 1897: 1881: 1878: 1877: 1861: 1858: 1857: 1841: 1838: 1837: 1820: 1816: 1814: 1811: 1810: 1786: 1782: 1777: 1774: 1773: 1770: 1742: 1706: 1702: 1690: 1677: 1673: 1661: 1655: 1652: 1651: 1645: 1638: 1613:, provided the 1566: 1562: 1550: 1544: 1541: 1540: 1508: 1504: 1492: 1486: 1483: 1482: 1459: 1455: 1450: 1447: 1446: 1425: 1423: 1415: 1412: 1411: 1392: 1388: 1387: 1383: 1371: 1359: 1356: 1355: 1335: 1331: 1326: 1323: 1322: 1320:natural numbers 1299: 1295: 1290: 1287: 1286: 1261: 1259: 1257: 1254: 1253: 1234: 1231: 1230: 1210: 1206: 1201: 1198: 1197: 1177: 1173: 1161: 1148: 1144: 1129: 1119: 1116: 1113: 1112: 1095: 1091: 1079: 1066: 1062: 1047: 1037: 1034: 1031: 1030: 976: 972: 947: 943: 931: 914: 901: 897: 885: 879: 876: 875: 858: 857: 851: 847: 835: 825: 824: 811: 798: 794: 782: 776: 773: 772: 770: 757: 756: 754:Similarly, the 701: 697: 672: 668: 656: 639: 626: 622: 610: 604: 601: 600: 583: 582: 576: 572: 560: 550: 549: 536: 523: 519: 507: 501: 498: 497: 495: 487:of a sequence ( 482: 481: 477: 453: 449: 434: 424: 417: 410: 406: 394: 388: 385: 384: 364: 360: 355: 352: 351: 331: 327: 312: 302: 295: 288: 284: 272: 266: 263: 262: 242: 238: 233: 230: 229: 211: 202: 131:, they are the 93: 82: 76: 73: 63:Please help to 62: 46: 42: 35: 28: 23: 22: 15: 12: 11: 5: 12390: 12380: 12379: 12365: 12364: 12344: 12343:External links 12341: 12339: 12338: 12332: 12319: 12313: 12299: 12296: 12295: 12290:Measure Theory 12280: 12250: 12215: 12190: 12161: 12143: 12114: 12096: 12089: 12068: 12067: 12065: 12062: 12061: 12060: 12055: 12050: 12045: 12040: 12033: 12030: 12028:respectively. 12017: 12006: 12005: 11993: 11990: 11986: 11982: 11977: 11973: 11969: 11966: 11963: 11960: 11955: 11951: 11947: 11942: 11938: 11934: 11931: 11928: 11925: 11902: 11881: 11877: 11874: 11854: 11851: 11846: 11842: 11821: 11816: 11812: 11808: 11788: 11768: 11757: 11756: 11744: 11741: 11738: 11735: 11730: 11726: 11722: 11719: 11716: 11713: 11708: 11704: 11700: 11695: 11691: 11687: 11684: 11681: 11678: 11655: 11635: 11632: 11629: 11609: 11606: 11601: 11597: 11572: 11568: 11564: 11561: 11558: 11536: 11533: 11530: 11526: 11520: 11516: 11512: 11492: 11481:limit superior 11468: 11465: 11461: 11460: 11449: 11446: 11443: 11440: 11435: 11431: 11427: 11422: 11418: 11414: 11411: 11407: 11404: 11401: 11398: 11397:lim inf 11380: 11379: 11368: 11365: 11362: 11357: 11353: 11349: 11344: 11338: 11335: 11329: 11325: 11321: 11318: 11315: 11312: 11311:lim inf 11300:is defined as 11294: 11293: 11282: 11279: 11276: 11273: 11268: 11264: 11260: 11255: 11251: 11247: 11244: 11240: 11237: 11234: 11231: 11230:lim sup 11217:order topology 11201: 11200: 11189: 11186: 11183: 11178: 11174: 11170: 11165: 11159: 11156: 11150: 11146: 11142: 11139: 11136: 11133: 11132:lim sup 11121:is defined as 11092: 11088: 11061: 11055: 11052: 11038: 11037: 11026: 11023: 11020: 11015: 11011: 11007: 11002: 10996: 10993: 10987: 10983: 10969:cluster points 10944: 10941: 10917: 10916: 10904: 10901: 10893: 10890: 10887: 10884: 10881: 10878: 10874: 10871: 10868: 10865: 10864:lim sup 10843: 10842: 10830: 10827: 10819: 10816: 10813: 10810: 10807: 10804: 10800: 10797: 10794: 10791: 10790:lim inf 10775:of all of the 10767: ⊆ 10760: 10757: 10752: 10749: 10748: 10747: 10746: 10745: 10719: 10718: 10711:dynamic system 10702: 10701: 10698: 10697: 10696: 10690: 10681: 10672: 10666: 10660: 10651: 10642: 10630: 10621: 10617:However, for ( 10615: 10614: 10613: 10607: 10601: 10595: 10583: 10576: 10575: 10574: 10563: 10560: 10557: 10554: 10551: 10548: 10544: 10540: 10537: 10534: 10531: 10528: 10524: 10520: 10517: 10514: 10511: 10508: 10504: 10500: 10497: 10494: 10491: 10488: 10484: 10480: 10477: 10474: 10471: 10468: 10464: 10460: 10457: 10454: 10451: 10448: 10444: 10440: 10437: 10434: 10431: 10428: 10425: 10422: 10419: 10416: 10413: 10410: 10407: 10404: 10399: 10395: 10391: 10378: 10377: 10369: 10368: 10364: 10363: 10343: 10342: 10341: 10335: 10329: 10323: 10314: 10313: 10312: 10301: 10298: 10295: 10292: 10289: 10286: 10283: 10280: 10277: 10274: 10271: 10268: 10265: 10262: 10259: 10256: 10253: 10250: 10247: 10244: 10241: 10238: 10235: 10232: 10229: 10226: 10223: 10220: 10217: 10214: 10211: 10208: 10205: 10202: 10199: 10196: 10193: 10190: 10187: 10184: 10181: 10178: 10175: 10172: 10169: 10166: 10163: 10160: 10155: 10151: 10147: 10134: 10133: 10125: 10124: 10123: 10122: 10116: 10107: 10098: 10092: 10086: 10077: 10068: 10056: 10047: 10043:However, for ( 10041: 10040: 10039: 10033: 10027: 10021: 10009: 10002: 10001: 10000: 9989: 9986: 9983: 9980: 9977: 9974: 9971: 9968: 9965: 9962: 9959: 9956: 9953: 9950: 9947: 9944: 9941: 9938: 9935: 9932: 9929: 9926: 9923: 9920: 9917: 9914: 9911: 9908: 9905: 9902: 9897: 9893: 9889: 9876: 9875: 9867: 9866: 9859: 9858: 9850: 9849: 9834: 9831: 9830: 9829: 9826: 9825: 9824: 9809: 9804: 9798: 9794: 9787: 9782: 9779: 9776: 9772: 9766: 9760: 9755: 9752: 9749: 9745: 9741: 9738: 9736: 9734: 9731: 9728: 9725: 9722: 9719: 9716: 9713: 9710: 9707: 9704: 9701: 9698: 9695: 9692: 9689: 9686: 9683: 9680: 9677: 9674: 9671: 9668: 9665: 9662: 9657: 9653: 9649: 9645: 9641: 9637: 9634: 9631: 9629: 9625: 9621: 9615: 9612: 9609: 9605: 9604:lim sup 9601: 9600: 9584: 9574: 9564: 9555: 9546: 9542:The sequence ( 9540: 9539: 9538: 9523: 9520: 9517: 9512: 9509: 9506: 9502: 9498: 9493: 9490: 9487: 9483: 9479: 9474: 9470: 9466: 9461: 9457: 9451: 9446: 9443: 9440: 9436: 9432: 9429: 9427: 9425: 9422: 9419: 9416: 9413: 9410: 9407: 9404: 9401: 9398: 9395: 9392: 9389: 9386: 9383: 9380: 9377: 9374: 9369: 9365: 9361: 9357: 9354: 9351: 9349: 9345: 9341: 9337: 9336: 9323: 9322: 9312: 9305: 9304: 9301: 9300: 9299: 9284: 9279: 9273: 9269: 9262: 9257: 9254: 9251: 9247: 9241: 9235: 9230: 9227: 9224: 9220: 9216: 9213: 9211: 9209: 9206: 9203: 9200: 9197: 9194: 9191: 9188: 9185: 9182: 9179: 9176: 9173: 9170: 9167: 9164: 9161: 9158: 9155: 9152: 9149: 9146: 9143: 9140: 9137: 9132: 9128: 9124: 9120: 9116: 9112: 9109: 9106: 9104: 9100: 9096: 9090: 9087: 9084: 9080: 9079:lim inf 9076: 9075: 9059: 9049: 9039: 9030: 9021: 9017:The sequence ( 9015: 9014: 9013: 8998: 8995: 8992: 8987: 8984: 8981: 8977: 8973: 8968: 8965: 8962: 8958: 8954: 8949: 8945: 8941: 8936: 8932: 8926: 8921: 8918: 8915: 8911: 8907: 8904: 8902: 8900: 8897: 8894: 8891: 8888: 8885: 8882: 8879: 8876: 8873: 8870: 8867: 8864: 8861: 8858: 8855: 8852: 8849: 8844: 8840: 8836: 8832: 8829: 8826: 8824: 8820: 8816: 8812: 8811: 8798: 8797: 8787: 8770: 8757: 8725: 8716: 8706: 8705: 8699: 8690: 8681: 8672: 8663: 8654: 8641: 8628: 8618: 8617: 8603: 8586: 8558: 8545: 8539: 8528: 8524: 8511: 8501: 8497: 8488: 8464: 8451: 8434: 8409: 8392: 8386: 8385: 8372: 8367: 8364: 8361: 8358: 8350: 8348: 8345: 8344: 8341: 8338: 8335: 8332: 8324: 8322: 8319: 8318: 8316: 8311: 8308: 8305: 8302: 8299: 8296: 8293: 8241:measure theory 8236: 8233: 8216: 8211: 8207: 8203: 8202:lim inf 8200: 8195: 8191: 8187: 8186:lim sup 8184: 8179: 8175: 8171: 8149: 8145: 8141: 8140:lim sup 8119: 8115: 8111: 8110:lim inf 8089: 8085: 8081: 8070: 8069: 8058: 8055: 8052: 8047: 8043: 8037: 8034: 8031: 8027: 8004: 8000: 7996: 7991: 7987: 7966: 7961: 7957: 7953: 7943: 7927: 7923: 7919: 7918:lim inf 7916: 7913: 7893: 7869: 7847: 7843: 7818: 7813: 7809: 7805: 7804:lim inf 7795: 7784: 7781: 7778: 7773: 7769: 7763: 7760: 7757: 7753: 7728: 7724: 7719: 7715: 7710: 7706: 7685: 7680: 7676: 7672: 7652: 7645: 7641: 7636: 7632: 7623: 7609: 7604: 7600: 7596: 7574: 7570: 7566: 7565:lim sup 7563: 7560: 7540: 7537: 7511: 7507: 7482: 7477: 7473: 7469: 7468:lim sup 7446: 7443: 7423: 7418: 7414: 7410: 7390: 7365: 7362: 7346: 7345: 7339: 7330: 7324: 7309: 7294: 7262:∅ because ∅ ⊆ 7220: 7217: 7188: 7187: 7176: 7173: 7170: 7167: 7164: 7161: 7158: 7155: 7152: 7149: 7145: 7142: 7139: 7136: 7132: 7128: 7125: 7122: 7119: 7112: 7109: 7106: 7103: 7100: 7097: 7094: 7091: 7088: 7085: 7082: 7079: 7076: 7073: 7070: 7067: 7064: 7061: 7057: 7053: 7049: 7046: 7043: 7040: 7037: 7034: 7029: 7026: 7023: 7019: 7018:lim inf 7008: 6997: 6994: 6991: 6988: 6985: 6982: 6979: 6976: 6973: 6970: 6966: 6963: 6960: 6957: 6953: 6949: 6946: 6943: 6940: 6933: 6930: 6927: 6924: 6921: 6918: 6915: 6912: 6909: 6906: 6903: 6900: 6897: 6894: 6891: 6888: 6885: 6882: 6878: 6874: 6870: 6867: 6864: 6861: 6858: 6855: 6850: 6847: 6844: 6840: 6839:lim sup 6800: 6797: 6785: 6781: 6777: 6774: 6771: 6768: 6765: 6762: 6759: 6756: 6753: 6750: 6747: 6744: 6741: 6738: 6735: 6732: 6729: 6726: 6723: 6720: 6717: 6713: 6709: 6703: 6700: 6697: 6693: 6689: 6686: 6683: 6680: 6677: 6672: 6669: 6666: 6662: 6661:lim inf 6652:and similarly 6640: 6636: 6633: 6630: 6627: 6624: 6621: 6618: 6615: 6612: 6609: 6606: 6603: 6600: 6597: 6594: 6591: 6588: 6585: 6582: 6579: 6576: 6572: 6568: 6562: 6559: 6556: 6552: 6548: 6545: 6542: 6539: 6536: 6531: 6528: 6525: 6521: 6520:lim sup 6507:non-increasing 6487: 6467: 6443: 6440: 6437: 6434: 6431: 6428: 6405: 6401: 6398: 6395: 6392: 6389: 6386: 6383: 6380: 6377: 6374: 6371: 6368: 6365: 6362: 6359: 6356: 6353: 6350: 6347: 6344: 6341: 6337: 6333: 6327: 6324: 6321: 6317: 6313: 6310: 6307: 6304: 6301: 6296: 6293: 6290: 6286: 6285:lim inf 6262: 6258: 6255: 6252: 6249: 6246: 6243: 6240: 6237: 6234: 6231: 6228: 6225: 6222: 6219: 6216: 6213: 6210: 6207: 6204: 6201: 6198: 6194: 6190: 6184: 6181: 6178: 6174: 6170: 6167: 6164: 6161: 6158: 6153: 6150: 6147: 6143: 6142:lim sup 6120: 6100: 6076: 6072: 6069: 6066: 6063: 6043: 6023: 6003: 5987: 5984: 5982: 5979: 5938: 5935: 5932: 5929: 5926: 5923: 5920: 5915: 5912: 5909: 5905: 5904:lim inf 5884: 5881: 5878: 5875: 5872: 5869: 5864: 5861: 5858: 5854: 5853:lim sup 5833: 5830: 5826: 5822: 5819: 5816: 5813: 5810: 5807: 5804: 5801: 5798: 5781: 5778: 5777: 5776: 5760: 5757: 5737: 5736: 5720: 5698: 5694: 5673: 5670: 5665: 5661: 5657: 5652: 5649: 5646: 5642: 5638: 5632: 5629: 5626: 5622: 5621:lim inf 5605: 5604: 5584: 5581: 5578: 5575: 5572: 5569: 5566: 5563: 5560: 5540: 5537: 5534: 5529: 5525: 5519: 5516: 5513: 5509: 5508:lim sup 5488: 5485: 5482: 5477: 5473: 5467: 5464: 5461: 5457: 5456:lim inf 5428: 5425: 5422: 5419: 5416: 5413: 5410: 5405: 5401: 5383: 5380: 5368: 5365: 5362: 5359: 5339: 5336: 5327:provided that 5316: 5313: 5310: 5306: 5300: 5296: 5290: 5286: 5281: 5275: 5272: 5269: 5265: 5264:lim sup 5244: 5239: 5235: 5229: 5226: 5223: 5219: 5218:lim sup 5215: 5212: 5192: 5189: 5186: 5183: 5163: 5160: 5155: 5151: 5145: 5142: 5139: 5135: 5112: 5109: 5106: 5103: 5092: 5091: 5079: 5073: 5069: 5063: 5060: 5057: 5053: 5052:lim inf 5048: 5041: 5035: 5031: 5025: 5022: 5019: 5015: 5014:lim inf 5010: 5006: 5003: 4998: 4994: 4988: 4984: 4980: 4974: 4971: 4968: 4964: 4963:lim inf 4942: 4935: 4931: 4925: 4922: 4919: 4915: 4914:lim sup 4910: 4903: 4896: 4892: 4886: 4883: 4880: 4876: 4875:lim sup 4871: 4867: 4864: 4859: 4855: 4849: 4845: 4841: 4835: 4832: 4829: 4825: 4824:lim sup 4804: 4801: 4796: 4792: 4788: 4785: 4782: 4777: 4773: 4769: 4745: 4723: 4719: 4713: 4710: 4707: 4703: 4702:lim inf 4680: 4676: 4670: 4667: 4664: 4660: 4659:lim sup 4639: 4636: 4633: 4628: 4624: 4603: 4598: 4594: 4588: 4585: 4582: 4578: 4577:lim inf 4571: 4566: 4562: 4556: 4553: 4550: 4546: 4545:lim inf 4542: 4539: 4534: 4530: 4526: 4521: 4517: 4513: 4507: 4504: 4501: 4497: 4496:lim inf 4481: 4480: 4469: 4464: 4460: 4454: 4451: 4448: 4444: 4443:lim sup 4437: 4432: 4428: 4422: 4419: 4416: 4412: 4411:lim sup 4408: 4405: 4400: 4396: 4392: 4387: 4383: 4379: 4373: 4370: 4367: 4363: 4362:lim sup 4342: 4339: 4336: 4333: 4313: 4310: 4307: 4283: 4280: 4275: 4271: 4267: 4264: 4261: 4256: 4252: 4248: 4233:cluster points 4220: 4215: 4211: 4205: 4201: 4197: 4192: 4188: 4182: 4179: 4176: 4172: 4171:lim sup 4168: 4163: 4159: 4153: 4150: 4147: 4143: 4142:lim inf 4139: 4134: 4130: 4124: 4120: 4108: 4107: 4095: 4075: 4053: 4049: 4028: 4008: 3986: 3982: 3970: 3958: 3938: 3916: 3912: 3891: 3871: 3849: 3845: 3829: 3828: 3816: 3813: 3791: 3787: 3766: 3755: 3743: 3740: 3718: 3714: 3693: 3667: 3664: 3659: 3655: 3649: 3646: 3643: 3639: 3638:lim sup 3635: 3630: 3626: 3622: 3619: 3616: 3611: 3607: 3601: 3598: 3595: 3591: 3590:lim inf 3569: 3565: 3561: 3558: 3537: 3533: 3528: 3524: 3501: 3498: 3493: 3489: 3483: 3480: 3477: 3473: 3472:lim sup 3469: 3462: 3458: 3453: 3436: 3432: 3427: 3423: 3420: 3417: 3412: 3408: 3402: 3399: 3396: 3392: 3391:lim inf 3369: 3365: 3342: 3338: 3315: 3311: 3286: 3282: 3277: 3252: 3248: 3243: 3219: 3216: 3213: 3210: 3207: 3187: 3184: 3162: 3158: 3137: 3134: 3131: 3128: 3108: 3105: 3102: 3099: 3096: 3093: 3090: 3087: 3084: 3081: 3061: 3056: 3052: 3031: 3028: 3025: 3022: 3019: 2997: 2993: 2987: 2984: 2981: 2977: 2976:lim sup 2973: 2970: 2948: 2944: 2938: 2935: 2932: 2928: 2927:lim inf 2924: 2921: 2895: 2892: 2889: 2886: 2881: 2877: 2871: 2868: 2865: 2861: 2848: 2845: 2842: 2839: 2836: 2834: 2830: 2826: 2820: 2817: 2814: 2810: 2809:lim sup 2806: 2805: 2802: 2799: 2796: 2791: 2787: 2781: 2778: 2775: 2771: 2758: 2755: 2752: 2749: 2747: 2743: 2739: 2733: 2730: 2727: 2723: 2722:lim inf 2719: 2718: 2698: 2678: 2675: 2655: 2651: 2628: 2624: 2618: 2615: 2612: 2608: 2598:in which case 2585: 2581: 2575: 2572: 2569: 2565: 2564:lim sup 2561: 2556: 2552: 2546: 2543: 2540: 2536: 2535:lim inf 2512: 2509: 2506: 2503: 2500: 2497: 2476: 2471: 2467: 2463: 2442: 2439: 2436: 2433: 2430: 2427: 2424: 2403: 2378: 2374: 2368: 2365: 2362: 2358: 2357:lim inf 2354: 2351: 2347: 2341: 2337: 2333: 2329: 2323: 2320: 2317: 2313: 2312:lim sup 2289: 2286: 2283: 2280: 2275: 2271: 2265: 2262: 2259: 2255: 2254:lim sup 2251: 2248: 2245: 2240: 2236: 2230: 2227: 2224: 2220: 2219:lim inf 2216: 2196: 2193: 2190: 2174: 2171: 2170: 2169: 2157: 2154: 2151: 2131: 2128: 2125: 2105: 2102: 2099: 2096: 2091: 2087: 2066: 2046: 2026: 2004: 2000: 1988: 1976: 1973: 1970: 1950: 1947: 1944: 1924: 1921: 1918: 1915: 1910: 1906: 1885: 1865: 1845: 1823: 1819: 1794: 1789: 1785: 1781: 1769: 1768:Interpretation 1766: 1741: 1738: 1730:big-O notation 1726: 1725: 1714: 1709: 1705: 1699: 1696: 1693: 1689: 1688:lim sup 1685: 1680: 1676: 1670: 1667: 1664: 1660: 1659:lim inf 1643: 1636: 1595: 1594: 1583: 1580: 1577: 1574: 1569: 1565: 1559: 1556: 1553: 1549: 1548:lim inf 1534: 1533: 1522: 1519: 1516: 1511: 1507: 1501: 1498: 1495: 1491: 1490:lim sup 1467: 1462: 1458: 1454: 1432: 1428: 1422: 1419: 1395: 1391: 1386: 1380: 1377: 1374: 1370: 1366: 1363: 1343: 1338: 1334: 1330: 1307: 1302: 1298: 1294: 1268: 1264: 1238: 1218: 1213: 1209: 1205: 1180: 1176: 1170: 1167: 1164: 1160: 1159:lim sup 1156: 1151: 1147: 1143: 1138: 1135: 1132: 1126: 1123: 1098: 1094: 1088: 1085: 1082: 1078: 1077:lim inf 1074: 1069: 1065: 1061: 1056: 1053: 1050: 1044: 1041: 1016: 1013: 1009: 1006: 1003: 1000: 997: 993: 990: 987: 984: 979: 975: 970: 966: 962: 958: 955: 950: 946: 940: 937: 934: 930: 923: 920: 917: 913: 909: 904: 900: 894: 891: 888: 884: 883:lim sup 861: 854: 850: 844: 841: 838: 834: 828: 820: 817: 814: 810: 806: 801: 797: 791: 788: 785: 781: 780:lim sup 766: 758:limit superior 741: 738: 734: 731: 728: 725: 722: 718: 715: 712: 709: 704: 700: 695: 691: 687: 683: 680: 675: 671: 665: 662: 659: 655: 648: 645: 642: 638: 634: 629: 625: 619: 616: 613: 609: 608:lim inf 586: 579: 575: 569: 566: 563: 559: 553: 545: 542: 539: 535: 531: 526: 522: 516: 513: 510: 506: 505:lim inf 491: 483:limit inferior 476: 473: 461: 456: 452: 448: 443: 440: 437: 431: 428: 413: 409: 403: 400: 397: 393: 392:lim sup 383:is denoted by 372: 367: 363: 359: 339: 334: 330: 326: 321: 318: 315: 309: 306: 291: 287: 281: 278: 275: 271: 270:lim inf 261:is denoted by 250: 245: 241: 237: 218:if and only if 207: 198: 177:superior limit 169:limit supremum 165:supremum limit 153:inferior limit 109:limit superior 105:limit inferior 95: 94: 49: 47: 40: 26: 9: 6: 4: 3: 2: 12389: 12378: 12375: 12374: 12372: 12361: 12357: 12356: 12351: 12347: 12346: 12335: 12333:0-8247-8415-4 12329: 12325: 12320: 12316: 12314:0-8176-7153-6 12310: 12306: 12301: 12300: 12291: 12284: 12276: 12272: 12268: 12264: 12257: 12255: 12240:on 2007-03-03 12236: 12232: 12225: 12219: 12204: 12203:Polymath wiki 12200: 12194: 12186: 12180: 12172: 12168: 12164: 12158: 12154: 12147: 12139: 12133: 12125: 12121: 12117: 12111: 12107: 12100: 12092: 12086: 12082: 12081: 12073: 12069: 12059: 12056: 12054: 12051: 12049: 12046: 12044: 12041: 12039: 12036: 12035: 12029: 12015: 11991: 11980: 11975: 11971: 11967: 11961: 11958: 11953: 11949: 11945: 11940: 11936: 11926: 11923: 11916: 11915: 11914: 11900: 11875: 11872: 11852: 11849: 11844: 11840: 11814: 11810: 11786: 11766: 11742: 11736: 11733: 11728: 11724: 11720: 11714: 11711: 11706: 11702: 11698: 11693: 11689: 11679: 11676: 11669: 11668: 11667: 11653: 11633: 11630: 11627: 11607: 11604: 11599: 11595: 11586: 11566: 11562: 11559: 11534: 11531: 11528: 11518: 11514: 11490: 11482: 11478: 11474: 11464: 11447: 11441: 11438: 11433: 11429: 11425: 11420: 11416: 11402: 11399: 11389: 11388: 11387: 11385: 11363: 11360: 11355: 11351: 11347: 11342: 11333: 11323: 11316: 11313: 11303: 11302: 11301: 11299: 11280: 11274: 11271: 11266: 11262: 11258: 11253: 11249: 11235: 11232: 11222: 11221: 11220: 11218: 11214: 11210: 11206: 11184: 11181: 11176: 11172: 11168: 11163: 11154: 11144: 11137: 11134: 11124: 11123: 11122: 11120: 11116: 11112: 11108: 11090: 11086: 11077: 11059: 11050: 11021: 11018: 11013: 11009: 11005: 11000: 10991: 10981: 10974: 10973: 10972: 10970: 10966: 10963: 10959: 10956: 10950: 10940: 10937: 10933: 10929: 10926: 10922: 10899: 10891: 10888: 10885: 10882: 10879: 10869: 10866: 10856: 10855: 10854: 10852: 10848: 10825: 10817: 10814: 10811: 10808: 10805: 10795: 10792: 10782: 10781: 10780: 10778: 10774: 10770: 10766: 10756: 10743: 10739: 10738:damping ratio 10735: 10731: 10727: 10723: 10722: 10721: 10720: 10716: 10712: 10708: 10704: 10703: 10699: 10693: 10689: 10684: 10680: 10675: 10671: 10667: 10663: 10659: 10654: 10650: 10645: 10641: 10637: 10636: 10633: 10629: 10624: 10620: 10616: 10610: 10606: 10602: 10598: 10594: 10590: 10589: 10586: 10582: 10577: 10561: 10555: 10552: 10546: 10542: 10538: 10532: 10526: 10522: 10518: 10512: 10506: 10502: 10498: 10492: 10486: 10482: 10478: 10472: 10466: 10462: 10458: 10452: 10446: 10442: 10438: 10432: 10426: 10420: 10414: 10405: 10397: 10393: 10382: 10381: 10380: 10379: 10375: 10371: 10370: 10366: 10365: 10361: 10357: 10353: 10349: 10344: 10338: 10334: 10330: 10326: 10322: 10318: 10317: 10315: 10299: 10293: 10290: 10284: 10278: 10272: 10266: 10260: 10254: 10248: 10242: 10236: 10230: 10224: 10218: 10212: 10209: 10203: 10197: 10194: 10188: 10182: 10176: 10170: 10161: 10153: 10149: 10138: 10137: 10136: 10135: 10131: 10127: 10126: 10119: 10115: 10110: 10106: 10101: 10097: 10093: 10089: 10085: 10080: 10076: 10071: 10067: 10063: 10062: 10059: 10055: 10050: 10046: 10042: 10036: 10032: 10028: 10024: 10020: 10016: 10015: 10012: 10008: 10003: 9987: 9981: 9978: 9972: 9966: 9960: 9954: 9948: 9942: 9936: 9930: 9924: 9918: 9912: 9903: 9895: 9891: 9880: 9879: 9878: 9877: 9873: 9869: 9868: 9865: 9861: 9860: 9856: 9852: 9851: 9848: 9844: 9843: 9842: 9840: 9827: 9807: 9802: 9796: 9792: 9780: 9777: 9774: 9770: 9764: 9753: 9750: 9747: 9743: 9739: 9737: 9723: 9720: 9717: 9714: 9711: 9705: 9702: 9699: 9690: 9687: 9684: 9681: 9678: 9675: 9672: 9666: 9663: 9660: 9655: 9651: 9632: 9630: 9623: 9619: 9607: 9591: 9590: 9587: 9583: 9577: 9573: 9567: 9563: 9558: 9554: 9549: 9545: 9541: 9521: 9518: 9515: 9510: 9507: 9504: 9500: 9496: 9491: 9488: 9485: 9481: 9477: 9472: 9468: 9464: 9459: 9455: 9444: 9441: 9438: 9434: 9430: 9428: 9414: 9411: 9408: 9405: 9402: 9399: 9396: 9393: 9390: 9387: 9384: 9378: 9375: 9372: 9367: 9363: 9352: 9350: 9343: 9339: 9327: 9326: 9325: 9324: 9320: 9315: 9311: 9307: 9306: 9302: 9282: 9277: 9271: 9267: 9255: 9252: 9249: 9245: 9239: 9228: 9225: 9222: 9218: 9214: 9212: 9198: 9195: 9192: 9189: 9186: 9180: 9177: 9174: 9165: 9162: 9159: 9156: 9153: 9150: 9147: 9141: 9138: 9135: 9130: 9126: 9107: 9105: 9098: 9094: 9082: 9066: 9065: 9062: 9058: 9052: 9048: 9042: 9038: 9033: 9029: 9024: 9020: 9016: 8996: 8993: 8990: 8985: 8982: 8979: 8975: 8971: 8966: 8963: 8960: 8956: 8952: 8947: 8943: 8939: 8934: 8930: 8919: 8916: 8913: 8909: 8905: 8903: 8889: 8886: 8883: 8880: 8877: 8874: 8871: 8868: 8865: 8862: 8859: 8853: 8850: 8847: 8842: 8838: 8827: 8825: 8818: 8814: 8802: 8801: 8800: 8799: 8795: 8790: 8786: 8782: 8781: 8780: 8778: 8773: 8769: 8765: 8760: 8756: 8752: 8748: 8744: 8740: 8736: 8735:set inclusion 8731: 8728: 8724: 8719: 8715: 8711: 8702: 8698: 8693: 8689: 8684: 8680: 8675: 8671: 8666: 8662: 8657: 8653: 8649: 8648: 8647: 8644: 8640: 8636: 8631: 8627: 8623: 8620:Observe that 8615: 8611: 8606: 8602: 8598: 8594: 8589: 8585: 8581: 8577: 8573: 8569: 8566: 8561: 8557: 8553: 8548: 8544: 8540: 8537: 8531: 8527: 8523: 8519: 8514: 8510: 8504: 8500: 8496: 8491: 8487: 8483: 8479: 8475: 8472: 8467: 8463: 8459: 8454: 8450: 8446: 8445: 8444: 8442: 8437: 8433: 8428: 8425: 8422:. Therefore, 8421: 8417: 8412: 8408: 8404: 8400: 8395: 8391: 8365: 8362: 8359: 8356: 8346: 8339: 8336: 8333: 8330: 8320: 8314: 8309: 8303: 8300: 8297: 8291: 8284: 8283: 8282: 8280: 8276: 8272: 8267: 8265: 8261: 8257: 8252: 8250: 8246: 8242: 8232: 8230: 8214: 8209: 8205: 8198: 8193: 8189: 8182: 8177: 8173: 8147: 8143: 8117: 8113: 8087: 8083: 8056: 8053: 8050: 8045: 8041: 8029: 8002: 7998: 7994: 7989: 7985: 7959: 7955: 7941: 7925: 7921: 7914: 7911: 7891: 7883: 7867: 7845: 7841: 7832: 7816: 7811: 7807: 7796: 7782: 7779: 7776: 7771: 7767: 7755: 7726: 7722: 7717: 7713: 7708: 7704: 7678: 7674: 7643: 7639: 7634: 7621: 7602: 7598: 7572: 7568: 7561: 7558: 7538: 7535: 7527: 7509: 7505: 7496: 7480: 7475: 7471: 7460: 7459: 7458: 7444: 7441: 7416: 7412: 7388: 7381: 7375: 7371: 7361: 7359: 7355: 7351: 7342: 7338: 7333: 7329: 7325: 7322: 7318: 7314: 7310: 7307: 7303: 7299: 7295: 7292: 7288: 7285:The sequence 7284: 7283: 7282: 7279: 7277: 7273: 7269: 7265: 7261: 7257: 7253: 7249: 7245: 7244:set inclusion 7241: 7237: 7234: 7230: 7226: 7216: 7214: 7210: 7206: 7201: 7197: 7193: 7168: 7162: 7153: 7150: 7147: 7143: 7140: 7137: 7134: 7130: 7110: 7107: 7098: 7089: 7086: 7083: 7080: 7077: 7074: 7068: 7062: 7044: 7038: 7032: 7027: 7021: 7009: 6989: 6983: 6974: 6971: 6968: 6964: 6961: 6958: 6955: 6951: 6931: 6928: 6919: 6910: 6907: 6904: 6901: 6898: 6895: 6889: 6883: 6865: 6859: 6853: 6848: 6842: 6830: 6829: 6828: 6826: 6825:neighborhoods 6822: 6818: 6814: 6810: 6806: 6796: 6783: 6779: 6769: 6757: 6754: 6751: 6745: 6742: 6739: 6736: 6733: 6730: 6724: 6718: 6707: 6701: 6698: 6695: 6687: 6681: 6675: 6670: 6664: 6638: 6628: 6616: 6613: 6610: 6604: 6601: 6598: 6595: 6592: 6589: 6583: 6577: 6566: 6560: 6557: 6554: 6546: 6540: 6534: 6529: 6523: 6510: 6508: 6504: 6501:Note that as 6499: 6485: 6465: 6457: 6438: 6435: 6432: 6426: 6417: 6403: 6393: 6381: 6378: 6375: 6369: 6366: 6363: 6360: 6357: 6354: 6348: 6342: 6331: 6325: 6319: 6311: 6305: 6299: 6294: 6288: 6275: 6260: 6250: 6238: 6235: 6232: 6226: 6223: 6220: 6217: 6214: 6211: 6205: 6199: 6188: 6182: 6176: 6168: 6162: 6156: 6151: 6145: 6132: 6118: 6098: 6091: 6067: 6064: 6061: 6041: 6034:contained in 6021: 6014:, a subspace 6001: 5993: 5978: 5976: 5975:badly behaved 5972: 5968: 5964: 5961:functions as 5960: 5956: 5952: 5936: 5933: 5930: 5924: 5918: 5913: 5907: 5882: 5879: 5873: 5867: 5862: 5856: 5828: 5824: 5820: 5814: 5811: 5808: 5802: 5796: 5787: 5774: 5755: 5747: 5743: 5739: 5738: 5734: 5718: 5696: 5692: 5671: 5663: 5659: 5655: 5650: 5647: 5644: 5640: 5624: 5611: 5610:number theory 5607: 5606: 5602: 5598: 5579: 5576: 5573: 5570: 5567: 5564: 5561: 5538: 5535: 5532: 5527: 5523: 5511: 5486: 5483: 5480: 5475: 5471: 5459: 5446: 5442: 5426: 5420: 5414: 5411: 5408: 5403: 5399: 5390: 5386: 5385: 5379: 5366: 5360: 5357: 5337: 5334: 5314: 5311: 5308: 5304: 5298: 5294: 5288: 5284: 5279: 5267: 5242: 5237: 5233: 5221: 5213: 5210: 5187: 5184: 5181: 5161: 5158: 5153: 5149: 5137: 5123: 5110: 5104: 5101: 5077: 5071: 5067: 5055: 5046: 5039: 5033: 5029: 5017: 5008: 5004: 4996: 4992: 4986: 4982: 4966: 4940: 4933: 4929: 4917: 4908: 4901: 4894: 4890: 4878: 4869: 4865: 4857: 4853: 4847: 4843: 4827: 4802: 4794: 4790: 4783: 4775: 4771: 4759: 4758: 4757: 4743: 4721: 4717: 4705: 4678: 4674: 4662: 4637: 4634: 4626: 4622: 4601: 4596: 4592: 4580: 4569: 4564: 4560: 4548: 4540: 4532: 4528: 4524: 4519: 4515: 4499: 4486: 4467: 4462: 4458: 4446: 4435: 4430: 4426: 4414: 4406: 4398: 4394: 4390: 4385: 4381: 4365: 4337: 4331: 4308: 4297: 4296:subadditivity 4281: 4273: 4269: 4262: 4254: 4250: 4238: 4237: 4236: 4234: 4218: 4213: 4209: 4203: 4195: 4190: 4186: 4174: 4166: 4161: 4157: 4145: 4137: 4132: 4128: 4122: 4093: 4073: 4051: 4047: 4026: 4006: 3984: 3980: 3971: 3929:greater than 3914: 3910: 3847: 3843: 3834: 3833: 3832: 3814: 3811: 3789: 3785: 3764: 3756: 3741: 3731:greater than 3716: 3712: 3683: 3682: 3681: 3678: 3665: 3662: 3657: 3653: 3641: 3633: 3628: 3624: 3620: 3617: 3614: 3609: 3605: 3593: 3567: 3563: 3559: 3556: 3531: 3526: 3522: 3512: 3499: 3496: 3491: 3487: 3475: 3467: 3460: 3456: 3451: 3434: 3430: 3425: 3421: 3418: 3415: 3410: 3406: 3394: 3367: 3363: 3340: 3336: 3313: 3309: 3284: 3280: 3275: 3250: 3246: 3241: 3233: 3214: 3211: 3208: 3185: 3182: 3160: 3156: 3148:will contain 3135: 3132: 3129: 3126: 3106: 3100: 3097: 3094: 3091: 3088: 3085: 3082: 3059: 3054: 3050: 3026: 3023: 3020: 2995: 2991: 2979: 2971: 2968: 2946: 2942: 2930: 2922: 2919: 2910: 2893: 2887: 2884: 2879: 2875: 2863: 2840: 2837: 2835: 2828: 2824: 2812: 2800: 2794: 2789: 2785: 2773: 2750: 2748: 2741: 2737: 2725: 2673: 2653: 2626: 2622: 2610: 2583: 2579: 2567: 2559: 2554: 2550: 2538: 2525: 2504: 2498: 2474: 2469: 2465: 2461: 2440: 2431: 2425: 2391: 2376: 2372: 2360: 2352: 2349: 2345: 2339: 2335: 2331: 2327: 2315: 2287: 2281: 2278: 2273: 2269: 2257: 2249: 2246: 2243: 2238: 2234: 2222: 2194: 2191: 2188: 2179: 2155: 2152: 2149: 2129: 2126: 2123: 2103: 2100: 2097: 2094: 2089: 2085: 2064: 2044: 2024: 2002: 1998: 1989: 1974: 1971: 1968: 1948: 1945: 1942: 1922: 1919: 1916: 1913: 1908: 1904: 1883: 1863: 1843: 1821: 1817: 1808: 1807: 1806: 1787: 1783: 1765: 1763: 1759: 1755: 1754:unbounded set 1751: 1747: 1737: 1734: 1731: 1712: 1707: 1703: 1691: 1683: 1678: 1674: 1662: 1650: 1649: 1648: 1646: 1639: 1632: 1626: 1624: 1620: 1616: 1612: 1608: 1604: 1600: 1581: 1578: 1572: 1567: 1563: 1551: 1539: 1538: 1537: 1520: 1514: 1509: 1505: 1493: 1481: 1480: 1479: 1460: 1456: 1420: 1417: 1393: 1389: 1384: 1372: 1364: 1361: 1336: 1332: 1321: 1300: 1296: 1284: 1252: 1236: 1229:. An element 1211: 1207: 1194: 1178: 1174: 1162: 1154: 1149: 1145: 1141: 1130: 1121: 1096: 1092: 1080: 1072: 1067: 1063: 1059: 1048: 1042: 1039: 1027: 1014: 1007: 1004: 1001: 998: 991: 988: 985: 982: 977: 973: 953: 948: 944: 938: 935: 932: 921: 918: 915: 907: 902: 898: 886: 852: 848: 842: 839: 836: 812: 804: 799: 795: 783: 769: 765: 761: 752: 739: 732: 729: 726: 723: 716: 713: 710: 707: 702: 698: 678: 673: 669: 663: 660: 657: 646: 643: 640: 632: 627: 623: 611: 577: 573: 567: 564: 561: 537: 529: 524: 520: 508: 494: 490: 486: 472: 459: 454: 450: 446: 435: 426: 411: 407: 395: 365: 361: 337: 332: 328: 324: 313: 307: 304: 289: 285: 273: 243: 239: 223: 219: 215: 210: 206: 201: 197: 192: 188: 186: 182: 178: 174: 170: 166: 162: 158: 154: 150: 146: 145:limit infimum 142: 141:infimum limit 138: 135:of the set's 134: 130: 126: 122: 118: 114: 110: 106: 102: 91: 88: 80: 77:February 2019 70: 66: 60: 59: 53: 48: 39: 38: 33: 19: 12353: 12323: 12304: 12289: 12283: 12269:(2): 28–93. 12266: 12262: 12242:. Retrieved 12235:the original 12230: 12218: 12206:. Retrieved 12202: 12193: 12152: 12146: 12105: 12099: 12079: 12072: 12007: 11758: 11585:directed set 11503:and the net 11470: 11462: 11383: 11381: 11297: 11295: 11215:and has the 11204: 11202: 11118: 11110: 11039: 10964: 10957: 10952: 10927: 10920: 10918: 10846: 10844: 10777:limit points 10768: 10764: 10762: 10754: 10736:(i.e., zero 10714: 10691: 10687: 10686:= lim 10682: 10678: 10673: 10669: 10661: 10657: 10656:= lim 10652: 10648: 10643: 10639: 10631: 10627: 10622: 10618: 10608: 10604: 10596: 10592: 10584: 10580: 10351: 10347: 10336: 10332: 10324: 10320: 10129: 10117: 10113: 10112:= lim 10108: 10104: 10099: 10095: 10087: 10083: 10082:= lim 10078: 10074: 10069: 10065: 10057: 10053: 10048: 10044: 10034: 10030: 10022: 10018: 10010: 10006: 9871: 9838: 9836: 9585: 9581: 9575: 9571: 9565: 9561: 9556: 9552: 9547: 9543: 9318: 9313: 9309: 9060: 9056: 9050: 9046: 9040: 9036: 9031: 9027: 9022: 9018: 8793: 8788: 8784: 8776: 8771: 8767: 8763: 8758: 8754: 8742: 8732: 8726: 8722: 8717: 8713: 8709: 8707: 8700: 8696: 8691: 8687: 8682: 8678: 8673: 8669: 8664: 8660: 8655: 8651: 8642: 8638: 8634: 8629: 8625: 8621: 8619: 8613: 8609: 8604: 8600: 8596: 8592: 8587: 8583: 8579: 8578:). That is, 8575: 8567: 8564: 8559: 8555: 8551: 8546: 8542: 8535: 8529: 8525: 8521: 8517: 8516:) such that 8512: 8508: 8502: 8498: 8494: 8489: 8485: 8481: 8480:). That is, 8473: 8470: 8465: 8461: 8457: 8452: 8448: 8440: 8435: 8431: 8429: 8423: 8419: 8415: 8410: 8406: 8402: 8398: 8393: 8389: 8387: 8278: 8274: 8270: 8268: 8259: 8255: 8253: 8238: 8071: 7904:). That is, 7830: 7494: 7377: 7357: 7347: 7340: 7336: 7331: 7327: 7321:contained in 7320: 7317:intersection 7305: 7290: 7286: 7280: 7275: 7271: 7267: 7263: 7255: 7251: 7247: 7235: 7228: 7222: 7212: 7204: 7189: 6820: 6816: 6812: 6808: 6802: 6511: 6502: 6500: 6454:denotes the 6418: 6276: 6133: 5992:metric space 5989: 5970: 5967:measure zero 5954: 5783: 5733:prime number 5124: 5093: 4482: 4109: 4066:lesser than 3830: 3679: 3513: 3232:subsequences 2911: 2392: 2302: 1771: 1750:real numbers 1743: 1735: 1727: 1641: 1634: 1630: 1627: 1599:real numbers 1596: 1535: 1282: 1195: 1028: 767: 763: 755: 753: 492: 488: 480: 478: 227: 213: 208: 204: 199: 195: 184: 180: 176: 172: 168: 164: 160: 156: 152: 148: 144: 140: 137:limit points 108: 104: 98: 83: 74: 55: 11913:defined by 11666:defined by 11209:total order 10962:filter base 10742:state space 10728:of several 8737:provides a 8570:(i.e., for 8245:probability 7831:inner limit 7622:subsequence 7524:taken from 7495:outer limit 7287:accumulates 6456:metric ball 6090:limit point 5951:oscillation 4110:In general, 214:accumulates 185:outer limit 181:upper limit 161:inner limit 157:lower limit 101:mathematics 69:introducing 12244:2006-02-24 12171:1074040561 12124:1074040561 12090:007054235X 12064:References 11113:is also a 11107:closed set 10947:See also: 10734:LTI system 9845:Using the 8650:lim 8572:cofinitely 8072:The limit 7977:such that 7944:of points 7882:cofinitely 7880:(that is, 7696:such that 7368:See also: 7211:, is 6458:of radius 5963:continuous 5844:, we have 5445:irrational 5391:function: 3804:less than 2173:Properties 2077:such that 1896:such that 1354:such that 222:convergent 52:references 12360:EMS Press 12179:cite book 12132:cite book 11981:∈ 11959:≤ 11876:∈ 11850:∈ 11734:∈ 11725:α 11715:α 11712:≤ 11703:α 11694:α 11631:∈ 11628:α 11605:∈ 11600:α 11567:≤ 11532:∈ 11529:α 11519:α 11477:sequences 11439:∈ 11361:∈ 11337:¯ 11324:⋂ 11272:∈ 11182:∈ 11158:¯ 11145:⋂ 11054:¯ 11019:∈ 10995:¯ 10982:⋂ 10883:∈ 10809:∈ 10707:limit set 10556:… 10352:essential 10294:… 10210:− 10195:− 9982:… 9786:∞ 9771:⋃ 9759:∞ 9744:⋂ 9724:… 9706:∈ 9691:… 9667:∈ 9614:∞ 9611:→ 9519:⋯ 9516:∪ 9497:∪ 9478:∪ 9450:∞ 9435:⋃ 9415:… 9379:∈ 9261:∞ 9246:⋂ 9234:∞ 9219:⋃ 9199:… 9181:∈ 9166:… 9142:∈ 9089:∞ 9086:→ 8994:⋯ 8991:∩ 8972:∩ 8953:∩ 8925:∞ 8910:⋂ 8890:… 8854:∈ 8775:, is the 8762:, is the 8360:≠ 8036:∞ 8033:→ 7995:∈ 7915:∈ 7762:∞ 7759:→ 7714:∈ 7562:∈ 7551:That is, 7260:empty set 7225:power set 7207:in , the 7172:∅ 7169:≠ 7157:∖ 7151:∩ 7138:∈ 7093:∖ 7087:∩ 7081:∈ 7025:→ 6993:∅ 6990:≠ 6978:∖ 6972:∩ 6959:∈ 6914:∖ 6908:∩ 6902:∈ 6846:→ 6764:∖ 6758:ε 6743:∩ 6737:∈ 6696:ε 6668:→ 6623:∖ 6617:ε 6602:∩ 6596:∈ 6555:ε 6527:→ 6466:ε 6439:ε 6388:∖ 6382:ε 6367:∩ 6361:∈ 6323:→ 6320:ε 6292:→ 6245:∖ 6239:ε 6224:∩ 6218:∈ 6180:→ 6177:ε 6149:→ 6071:→ 5934:− 5911:→ 5860:→ 5815:⁡ 5759:∞ 5656:− 5631:∞ 5628:→ 5580:… 5518:∞ 5515:→ 5484:− 5466:∞ 5463:→ 5415:⁡ 5364:∞ 5361:⋅ 5274:∞ 5271:→ 5228:∞ 5225:→ 5191:∞ 5144:∞ 5141:→ 5108:∞ 5105:⋅ 5062:∞ 5059:→ 5024:∞ 5021:→ 5005:≥ 4973:∞ 4970:→ 4924:∞ 4921:→ 4885:∞ 4882:→ 4866:≤ 4834:∞ 4831:→ 4712:∞ 4709:→ 4669:∞ 4666:→ 4632:→ 4587:∞ 4584:→ 4555:∞ 4552:→ 4541:≥ 4506:∞ 4503:→ 4453:∞ 4450:→ 4421:∞ 4418:→ 4407:≤ 4372:∞ 4369:→ 4341:∞ 4335:∞ 4332:− 4312:∞ 4309:− 4306:∞ 4196:≤ 4181:∞ 4178:→ 4167:≤ 4152:∞ 4149:→ 4138:≤ 4094:λ 4074:λ 4027:λ 4007:λ 3957:Λ 3937:Λ 3890:Λ 3870:Λ 3812:λ 3765:λ 3739:Λ 3692:Λ 3666:ϵ 3648:∞ 3645:→ 3618:ϵ 3615:− 3600:∞ 3597:→ 3560:≥ 3532:∈ 3500:ϵ 3497:− 3482:∞ 3479:→ 3419:ϵ 3401:∞ 3398:→ 3127:ϵ 3101:ϵ 3089:ϵ 3086:− 2986:∞ 2983:→ 2937:∞ 2934:→ 2891:∞ 2888:− 2870:∞ 2867:→ 2844:∞ 2841:− 2819:∞ 2816:→ 2798:∞ 2780:∞ 2777:→ 2754:∞ 2732:∞ 2729:→ 2697:∞ 2677:∞ 2674:− 2617:∞ 2614:→ 2574:∞ 2571:→ 2545:∞ 2542:→ 2524:converges 2508:∞ 2502:∞ 2499:− 2435:∞ 2429:∞ 2426:− 2367:∞ 2364:→ 2353:− 2332:− 2322:∞ 2319:→ 2282:ϵ 2264:∞ 2261:→ 2247:ϵ 2244:− 2229:∞ 2226:→ 2189:ϵ 2156:ε 2153:− 2104:ε 2101:− 2045:ε 1975:ε 1923:ε 1864:ε 1698:∞ 1695:→ 1684:≤ 1669:∞ 1666:→ 1558:∞ 1555:→ 1500:∞ 1497:→ 1431:¯ 1421:⊆ 1379:∞ 1376:→ 1362:ξ 1267:¯ 1237:ξ 1169:∞ 1166:→ 1142:⁡ 1137:∞ 1134:→ 1125:¯ 1087:∞ 1084:→ 1060:⁡ 1055:∞ 1052:→ 1043:_ 1005:≥ 989:≥ 936:≥ 919:≥ 893:∞ 890:→ 840:≥ 819:∞ 816:→ 790:∞ 787:→ 730:≥ 714:≥ 661:≥ 644:≥ 618:∞ 615:→ 565:≥ 544:∞ 541:→ 515:∞ 512:→ 447:⁡ 442:∞ 439:→ 430:¯ 402:∞ 399:→ 325:⁡ 320:∞ 317:→ 308:_ 280:∞ 277:→ 127:). For a 12371:Category 12305:Analysis 12032:See also 11865:for any 11620:for all 11549:, where 10851:supremum 10715:converge 9833:Examples 9035:⊆ 8608:for all 8534:for all 8353:if 8327:if 7942:sequence 7350:topology 7306:contains 7194:and the 5382:Examples 2116:for all 1935:for all 1607:complete 1478:, then 121:function 117:limiting 113:sequence 12362:, 2001 11211:, is a 11076:closure 11074:is the 10953:Take a 10849:is the 10773:infimum 10771:is the 10600:= {0,1} 10328:= {0,1} 10026:= {0,1} 7231:) of a 6807:. Take 5711:is the 4086:, then 4019:, then 3949:, then 3882:, then 3328:(where 1615:suprema 1249:of the 65:improve 12330: 12311: 12208:14 May 12169: 12159: 12122: 12112: 12087: 11832:where 11207:has a 11040:where 10960:and a 10730:stable 8507:) of ( 7620:and a 7457:then: 7114: 6935: 6478:about 6419:where 5786:subset 5746:proven 5684:where 5441:π 5203:) and 4573: 4439: 2453:Then, 1619:infima 1605:) are 173:limsup 149:liminf 103:, the 54:, but 12238:(PDF) 12227:(PDF) 11583:is a 10695:= {0} 10665:= {1} 10612:= { } 10348:tails 10340:= { } 10121:= {1} 10091:= {0} 10038:= { } 8612:> 8574:many 8476:(see 7884:many 7528:many 7302:union 7298:joins 7238:is a 5255:then 1536:and 1410:. If 1281:is a 183:, or 159:, or 123:(see 111:of a 12328:ISBN 12309:ISBN 12210:2014 12185:link 12167:OCLC 12157:ISBN 12138:link 12120:OCLC 12110:ISBN 12085:ISBN 11587:and 11473:nets 10358:and 9853:The 8783:Let 8751:join 8747:meet 8563:for 8469:for 8430:If ( 8243:and 8132:and 8017:and 7743:and 7372:and 7313:meet 7291:sets 7223:The 7192:nets 6815:and 6699:> 6558:> 6274:and 5973:is " 5895:and 5731:-th 5499:and 5389:sine 4954:and 4756:). 3634:< 3621:< 3468:> 3422:> 3355:and 3267:and 3130:> 2961:and 2192:> 2127:> 2095:> 1946:> 1914:< 1617:and 1111:and 762:of ( 479:The 107:and 12271:doi 11413:inf 11406:sup 11320:inf 11246:sup 11239:inf 11141:sup 11078:of 10873:sup 10799:inf 10198:100 9644:sup 9636:inf 9356:sup 9119:inf 9111:sup 8831:inf 8256:and 8170:lim 8080:lim 8026:lim 7752:lim 7663:of 7278:). 7250:of 7233:set 7056:inf 7048:sup 6877:sup 6869:inf 6712:inf 6692:sup 6571:sup 6551:inf 6336:inf 6316:lim 6193:sup 6173:lim 6111:of 5953:of 5812:sin 5612:is 5595:is 5443:is 5412:sin 5134:lim 5125:If 4693:or 4353:): 4324:or 4235:. 4200:sup 4119:inf 3757:If 3684:If 3301:of 2912:If 2860:lim 2770:lim 2689:or 2607:lim 2488:in 2415:to 1744:In 1631:not 1576:inf 1518:sup 1369:lim 1285:of 1122:lim 1040:lim 965:sup 957:inf 929:sup 912:inf 874:or 833:sup 809:lim 690:inf 682:sup 654:inf 637:sup 599:or 558:inf 534:lim 427:lim 305:lim 129:set 99:In 12373:: 12358:, 12352:, 12267:29 12265:. 12253:^ 12229:. 12201:. 12181:}} 12177:{{ 12165:. 12134:}} 12130:{{ 12118:. 11927::= 11680::= 11317::= 11219:, 11138::= 10870::= 10796::= 10213:25 10183:20 10171:50 9841:. 9578:+1 9568:+1 9560:⊇ 9053:+1 9043:+1 8730:. 8646:. 8599:∈ 8520:∈ 8414:= 8401:∈ 8310::= 8277:∈ 8273:, 8266:. 8251:. 7360:. 7266:⊆ 7227:℘( 6827:: 6811:, 6498:. 6131:, 5603:.) 5539:1. 1625:. 1155::= 1073::= 908::= 805::= 633::= 530::= 419:or 297:or 187:. 179:, 175:, 171:, 167:, 155:, 151:, 147:, 143:, 12336:. 12317:. 12277:. 12273:: 12247:. 12212:. 12187:) 12173:. 12140:) 12126:. 12093:. 12016:C 11992:. 11989:} 11985:N 11976:0 11972:n 11968:: 11965:} 11962:n 11954:0 11950:n 11946:: 11941:n 11937:x 11933:{ 11930:{ 11924:C 11901:C 11880:N 11873:n 11853:X 11845:n 11841:x 11820:) 11815:n 11811:x 11807:( 11787:X 11767:B 11743:. 11740:} 11737:A 11729:0 11721:: 11718:} 11707:0 11699:: 11690:x 11686:{ 11683:{ 11677:B 11654:B 11634:A 11608:X 11596:x 11571:) 11563:, 11560:A 11557:( 11535:A 11525:) 11515:x 11511:( 11491:X 11448:. 11445:} 11442:B 11434:0 11430:B 11426:: 11421:0 11417:B 11410:{ 11403:= 11400:B 11384:X 11367:} 11364:B 11356:0 11352:B 11348:: 11343:0 11334:B 11328:{ 11314:B 11298:B 11281:. 11278:} 11275:B 11267:0 11263:B 11259:: 11254:0 11250:B 11243:{ 11236:= 11233:B 11205:X 11188:} 11185:B 11177:0 11173:B 11169:: 11164:0 11155:B 11149:{ 11135:B 11119:B 11111:X 11091:0 11087:B 11060:0 11051:B 11025:} 11022:B 11014:0 11010:B 11006:: 11001:0 10992:B 10986:{ 10965:B 10958:X 10928:Y 10921:X 10903:} 10900:X 10892:x 10889:: 10886:Y 10880:x 10877:{ 10867:X 10847:X 10829:} 10826:X 10818:x 10815:: 10812:Y 10806:x 10803:{ 10793:X 10769:Y 10765:X 10692:n 10688:Z 10683:n 10679:Z 10674:n 10670:Z 10662:n 10658:Y 10653:n 10649:Y 10644:n 10640:Y 10632:n 10628:Z 10623:n 10619:Y 10609:n 10605:X 10597:n 10593:X 10585:n 10581:X 10562:. 10559:) 10553:, 10550:} 10547:4 10543:/ 10539:1 10536:{ 10533:, 10530:} 10527:4 10523:/ 10519:3 10516:{ 10513:, 10510:} 10507:3 10503:/ 10499:1 10496:{ 10493:, 10490:} 10487:3 10483:/ 10479:2 10476:{ 10473:, 10470:} 10467:2 10463:/ 10459:1 10456:{ 10453:, 10450:} 10447:2 10443:/ 10439:1 10436:{ 10433:, 10430:} 10427:1 10424:{ 10421:, 10418:} 10415:0 10412:{ 10409:( 10406:= 10403:) 10398:n 10394:X 10390:( 10376:: 10337:n 10333:X 10325:n 10321:X 10300:. 10297:) 10291:, 10288:} 10285:1 10282:{ 10279:, 10276:} 10273:0 10270:{ 10267:, 10264:} 10261:1 10258:{ 10255:, 10252:} 10249:0 10246:{ 10243:, 10240:} 10237:1 10234:{ 10231:, 10228:} 10225:0 10222:{ 10219:, 10216:} 10207:{ 10204:, 10201:} 10192:{ 10189:, 10186:} 10180:{ 10177:, 10174:} 10168:{ 10165:( 10162:= 10159:) 10154:n 10150:X 10146:( 10130:X 10118:n 10114:Z 10109:n 10105:Z 10100:n 10096:Z 10088:n 10084:Y 10079:n 10075:Y 10070:n 10066:Y 10058:n 10054:Z 10049:n 10045:Y 10035:n 10031:X 10023:n 10019:X 10011:n 10007:X 10005:( 9988:. 9985:) 9979:, 9976:} 9973:1 9970:{ 9967:, 9964:} 9961:0 9958:{ 9955:, 9952:} 9949:1 9946:{ 9943:, 9940:} 9937:0 9934:{ 9931:, 9928:} 9925:1 9922:{ 9919:, 9916:} 9913:0 9910:{ 9907:( 9904:= 9901:) 9896:n 9892:X 9888:( 9872:X 9839:X 9808:. 9803:) 9797:m 9793:X 9781:n 9778:= 9775:m 9765:( 9754:1 9751:= 9748:n 9740:= 9730:} 9727:} 9721:, 9718:2 9715:, 9712:1 9709:{ 9703:n 9700:: 9697:} 9694:} 9688:, 9685:1 9682:+ 9679:n 9676:, 9673:n 9670:{ 9664:m 9661:: 9656:m 9652:X 9648:{ 9640:{ 9633:= 9624:n 9620:X 9608:n 9586:n 9582:J 9576:n 9572:J 9566:n 9562:J 9557:n 9553:J 9548:n 9544:J 9522:. 9511:2 9508:+ 9505:n 9501:X 9492:1 9489:+ 9486:n 9482:X 9473:n 9469:X 9465:= 9460:m 9456:X 9445:n 9442:= 9439:m 9431:= 9421:} 9418:} 9412:, 9409:2 9406:+ 9403:n 9400:, 9397:1 9394:+ 9391:n 9388:, 9385:n 9382:{ 9376:m 9373:: 9368:m 9364:X 9360:{ 9353:= 9344:n 9340:J 9319:n 9314:n 9310:J 9283:. 9278:) 9272:m 9268:X 9256:n 9253:= 9250:m 9240:( 9229:1 9226:= 9223:n 9215:= 9205:} 9202:} 9196:, 9193:2 9190:, 9187:1 9184:{ 9178:n 9175:: 9172:} 9169:} 9163:, 9160:1 9157:+ 9154:n 9151:, 9148:n 9145:{ 9139:m 9136:: 9131:m 9127:X 9123:{ 9115:{ 9108:= 9099:n 9095:X 9083:n 9061:n 9057:I 9051:n 9047:I 9041:n 9037:I 9032:n 9028:I 9023:n 9019:I 8997:. 8986:2 8983:+ 8980:n 8976:X 8967:1 8964:+ 8961:n 8957:X 8948:n 8944:X 8940:= 8935:m 8931:X 8920:n 8917:= 8914:m 8906:= 8896:} 8893:} 8887:, 8884:2 8881:+ 8878:n 8875:, 8872:1 8869:+ 8866:n 8863:, 8860:n 8857:{ 8851:m 8848:: 8843:m 8839:X 8835:{ 8828:= 8819:n 8815:I 8794:n 8789:n 8785:I 8772:n 8768:X 8759:n 8755:X 8743:X 8727:n 8723:X 8718:n 8714:X 8710:X 8704:. 8701:n 8697:X 8692:n 8688:X 8683:n 8679:X 8674:n 8670:X 8665:n 8661:X 8656:n 8652:X 8643:n 8639:X 8635:x 8630:n 8626:X 8622:x 8616:. 8614:m 8610:n 8605:n 8601:X 8597:x 8593:m 8588:n 8584:X 8580:x 8576:n 8568:n 8560:n 8556:X 8552:X 8547:n 8543:X 8538:. 8536:k 8530:k 8526:n 8522:X 8518:x 8513:n 8509:X 8503:k 8499:n 8495:X 8490:n 8486:X 8482:x 8474:n 8466:n 8462:X 8458:X 8453:n 8449:X 8441:X 8436:n 8432:X 8420:k 8416:x 8411:k 8407:x 8403:X 8399:x 8394:k 8390:x 8366:, 8363:y 8357:x 8347:1 8340:, 8337:y 8334:= 8331:x 8321:0 8315:{ 8307:) 8304:y 8301:, 8298:x 8295:( 8292:d 8279:X 8275:y 8271:x 8260:X 8215:. 8210:n 8206:X 8199:= 8194:n 8190:X 8183:= 8178:n 8174:X 8148:n 8144:X 8118:n 8114:X 8088:n 8084:X 8057:. 8054:x 8051:= 8046:k 8042:x 8030:k 8003:k 7999:X 7990:k 7986:x 7965:) 7960:k 7956:x 7952:( 7926:n 7922:X 7912:x 7892:n 7868:n 7846:n 7842:X 7817:, 7812:n 7808:X 7783:. 7780:x 7777:= 7772:k 7768:x 7756:k 7727:k 7723:n 7718:X 7709:k 7705:x 7684:) 7679:n 7675:X 7671:( 7651:) 7644:k 7640:n 7635:X 7631:( 7608:) 7603:k 7599:x 7595:( 7573:n 7569:X 7559:x 7539:. 7536:n 7510:n 7506:X 7481:, 7476:n 7472:X 7445:, 7442:X 7422:) 7417:n 7413:X 7409:( 7389:X 7358:X 7341:n 7337:X 7332:n 7328:X 7276:X 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5315:B 5312:A 5309:= 5305:) 5299:n 5295:b 5289:n 5285:a 5280:( 5268:n 5243:, 5238:n 5234:b 5222:n 5214:= 5211:B 5188:+ 5185:= 5182:A 5162:A 5159:= 5154:n 5150:a 5138:n 5111:. 5102:0 5078:) 5072:n 5068:b 5056:n 5047:( 5040:) 5034:n 5030:a 5018:n 5009:( 5002:) 4997:n 4993:b 4987:n 4983:a 4979:( 4967:n 4941:) 4934:n 4930:b 4918:n 4909:( 4902:) 4895:n 4891:a 4879:n 4870:( 4863:) 4858:n 4854:b 4848:n 4844:a 4840:( 4828:n 4803:, 4800:) 4795:n 4791:b 4787:( 4784:, 4781:) 4776:n 4772:a 4768:( 4744:a 4722:n 4718:a 4706:n 4679:n 4675:a 4663:n 4638:, 4635:a 4627:n 4623:a 4602:. 4597:n 4593:b 4581:n 4570:+ 4565:n 4561:a 4549:n 4538:) 4533:n 4529:b 4525:+ 4520:n 4516:a 4512:( 4500:n 4487:: 4468:. 4463:n 4459:b 4447:n 4436:+ 4431:n 4427:a 4415:n 4404:) 4399:n 4395:b 4391:+ 4386:n 4382:a 4378:( 4366:n 4338:+ 4282:, 4279:) 4274:n 4270:b 4266:( 4263:, 4260:) 4255:n 4251:a 4247:( 4219:. 4214:n 4210:x 4204:n 4191:n 4187:x 4175:n 4162:n 4158:x 4146:n 4133:n 4129:x 4123:n 4052:n 4048:x 3985:n 3981:x 3915:n 3911:x 3848:n 3844:x 3815:; 3790:n 3786:x 3742:; 3717:n 3713:x 3663:+ 3658:n 3654:x 3642:n 3629:n 3625:x 3610:n 3606:x 3594:n 3568:0 3564:n 3557:n 3536:N 3527:0 3523:n 3492:n 3488:x 3476:n 3461:n 3457:k 3452:x 3435:n 3431:h 3426:x 3416:+ 3411:n 3407:x 3395:n 3368:n 3364:h 3341:n 3337:k 3314:n 3310:x 3285:n 3281:h 3276:x 3251:n 3247:k 3242:x 3218:] 3215:S 3212:, 3209:I 3206:[ 3186:. 3183:n 3161:n 3157:x 3136:, 3133:0 3107:, 3104:] 3098:+ 3095:S 3092:, 3083:I 3080:[ 3060:, 3055:n 3051:x 3030:] 3027:S 3024:, 3021:I 3018:[ 2996:n 2992:x 2980:n 2972:= 2969:S 2947:n 2943:x 2931:n 2923:= 2920:I 2894:. 2885:= 2880:n 2876:x 2864:n 2838:= 2829:n 2825:x 2813:n 2801:, 2795:= 2790:n 2786:x 2774:n 2751:= 2742:n 2738:x 2726:n 2654:, 2650:R 2627:n 2623:x 2611:n 2584:n 2580:x 2568:n 2560:= 2555:n 2551:x 2539:n 2511:] 2505:, 2496:[ 2475:) 2470:n 2466:x 2462:( 2441:. 2438:] 2432:, 2423:[ 2402:R 2377:n 2373:x 2361:n 2350:= 2346:) 2340:n 2336:x 2328:( 2316:n 2288:. 2285:) 2279:+ 2274:n 2270:x 2258:n 2250:, 2239:n 2235:x 2223:n 2215:( 2195:0 2168:. 2150:b 2130:N 2124:n 2098:b 2090:n 2086:x 2065:N 2025:b 2003:n 1999:x 1987:. 1972:+ 1969:b 1949:N 1943:n 1920:+ 1917:b 1909:n 1905:x 1884:N 1844:b 1822:n 1818:x 1793:) 1788:n 1784:x 1780:( 1713:. 1708:n 1704:x 1692:n 1679:n 1675:x 1663:n 1644:n 1642:x 1637:n 1635:x 1582:. 1579:E 1573:= 1568:n 1564:x 1552:n 1521:E 1515:= 1510:n 1506:x 1494:n 1466:) 1461:n 1457:x 1453:( 1427:R 1418:E 1394:k 1390:n 1385:x 1373:k 1365:= 1342:) 1337:k 1333:n 1329:( 1306:) 1301:n 1297:x 1293:( 1263:R 1217:) 1212:n 1208:x 1204:( 1179:n 1175:x 1163:n 1150:n 1146:x 1131:n 1097:n 1093:x 1081:n 1068:n 1064:x 1049:n 1015:. 1012:} 1008:0 1002:n 999:: 996:} 992:n 986:m 983:: 978:m 974:x 969:{ 961:{ 954:= 949:m 945:x 939:n 933:m 922:0 916:n 903:n 899:x 887:n 860:) 853:m 849:x 843:n 837:m 827:( 813:n 800:n 796:x 784:n 768:n 764:x 740:. 737:} 733:0 727:n 724:: 721:} 717:n 711:m 708:: 703:m 699:x 694:{ 686:{ 679:= 674:m 670:x 664:n 658:m 647:0 641:n 628:n 624:x 612:n 585:) 578:m 574:x 568:n 562:m 552:( 538:n 525:n 521:x 509:n 493:n 489:x 460:. 455:n 451:x 436:n 412:n 408:x 396:n 371:) 366:n 362:x 358:( 338:, 333:n 329:x 314:n 290:n 286:x 274:n 249:) 244:n 240:x 236:( 209:n 205:x 200:n 196:x 90:) 84:( 79:) 75:( 61:. 34:. 20:)

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Knowledge

Limit inferior and limit superior

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