2798:
2924:
2602:
2885:
2702:
2759:
3011:
2797:
2923:
2601:
4663:
of consumption bundles for agents in an economy, and an allocation is Pareto efficient if no change can be made to it that makes no agent worse off and at least one agent better off (here rows of the allocation matrix must be rankable by a
619:
2884:
2701:
2758:
513:
1249:
3010:
1834:
2234:
3810:
3685:
2975:
2849:
2653:
1169:
1021:
4393:
2290:
2152:
2486:
4038:
4633:
4563:
3905:
3479:
2548:
2096:
1747:
1665:
1603:
1511:
1482:
1420:
1391:
1347:
1311:
681:
328:
260:
227:
146:
106:
73:
4118:
3162:
409:
4318:
4010:
2433:
2027:
1914:
1115:
967:
1537:
807:
721:
4357:
4151:
3724:
846:
2389:
2326:
2188:
2063:
1983:
1870:
3618:
3585:
3532:
3446:
3296:
2593:
2519:
1947:
1718:
1636:
1453:
1282:
652:
299:
3116:
2750:
435:
4065:
3413:
3366:
3239:
3192:
3053:
3002:
2915:
2876:
2789:
2353:
1075:
1048:
927:
900:
873:
767:
188:. Some fifty years later the result was identified as significant in its own right, and proved again by Weierstrass. It has since become an essential theorem of
4588:
4533:
4508:
4486:
4460:
4438:
4413:
4277:
4242:
4220:
4198:
4172:
3970:
3948:
3926:
3875:
3836:
3768:
3746:
3638:
3552:
3499:
3386:
3339:
3319:
3259:
3212:
2693:
2673:
1685:
1557:
741:
369:
518:
4655:
concepts in economics, the proofs of the existence of which often require variations of the
Bolzano–Weierstrass theorem. One example is the existence of a
2803:
Because each sequence has infinitely many members, there must be (at least) one of these subintervals that contains infinitely many members of
4862:
440:
4643:
is compact if and only if it is sequentially compact, so that the
Bolzano–Weierstrass and Heine–Borel theorems are essentially the same.
1174:
437:. Suppose first that the sequence has infinitely many peaks, which means there is a subsequence with the following indices
1752:
4696:
4803:
4784:
2193:
4691:
3021:
Because we halve the length of an interval at each step, the limit of the interval's length is zero. Also, by the
4820:
3780:
3655:
2932:
2806:
2610:
4867:
4830:
1120:
972:
4363:
2239:
2101:
1354:
40:
2438:
4825:
4322:
4681:
4015:
185:
149:
4609:
4539:
3881:
3451:
2524:
2068:
1723:
1641:
1562:
1487:
1458:
1396:
1367:
1323:
1287:
657:
304:
236:
203:
122:
82:
49:
4668:). The Bolzano–Weierstrass theorem allows one to prove that if the set of allocations is compact and
4070:
3127:
374:
4284:
3976:
2394:
1988:
1875:
1080:
932:
1516:
4686:
4603:
772:
686:
4329:
4123:
3696:
812:
2358:
2295:
2157:
2032:
1952:
1839:
3590:
3557:
3504:
3418:
3268:
2565:
2491:
2328:
has a convergent subsequence. This reasoning may be applied until we obtain a countable set
1919:
1690:
1608:
1425:
1361:
1254:
624:
271:
3061:
2710:
414:
4660:
4652:
4592:
4252:
4043:
3850:
3814:
3689:
3391:
3344:
3217:
3170:
3031:
3016:
We continue this process infinitely many times. Thus we get a sequence of nested intervals.
2980:
2893:
2854:
2767:
2331:
1053:
1026:
905:
878:
851:
4842:
8:
4665:
746:
614:{\displaystyle x_{n_{1}}\geq x_{n_{2}}\geq x_{n_{3}}\geq \dots \geq x_{n_{j}}\geq \dots }
181:
109:
4773:
4573:
4518:
4493:
4471:
4445:
4423:
4416:
4398:
4262:
4227:
4205:
4183:
4176:
4157:
3955:
3933:
3911:
3860:
3821:
3753:
3731:
3623:
3537:
3484:
3371:
3324:
3304:
3262:
3244:
3197:
2678:
2658:
1670:
1542:
726:
354:
4799:
4780:
4656:
4596:
331:
4639:
if and only if it is closed and bounded. In fact, general topology tells us that a
4768:
4640:
3023:
2559:
1317:
177:
76:
36:
16:
Bounded sequence in finite-dimensional
Euclidean space has a convergent subsequence
173:
44:
32:
1171:. Repeating this process leads to an infinite non-decreasing subsequence
342:
338:
4847:
3341:. Because the length of the intervals converges to zero, there is an interval
4856:
4636:
189:
24:
2558:
There is also an alternative proof of the
Bolzano–Weierstrass theorem using
1513:) has a convergent subsequence if and only if there exists a countable set
1350:
334:
230:
157:
112:
20:
4464:
3840:
153:
4669:
2929:
Again, one of these subintervals contains infinitely many members of
2065:
has a convergent subsequence and hence there exists a countable set
508:{\displaystyle n_{1}<n_{2}<n_{3}<\dots <n_{j}<\dots }
4602:
This form of the theorem makes especially clear the analogy to the
1353:
a monotone subsequence, likewise also bounded. It follows from the
3973:
must be bounded, since otherwise the following unbounded sequence
1244:{\displaystyle x_{n_{1}}\leq x_{n_{2}}\leq x_{n_{3}}\leq \ldots }
2190:, by applying the lemma once again there exists a countable set
4837:
723:. But suppose now that there are only finitely many peaks, let
262:
can be put to good use. Indeed, we have the following result:
116:
3167:
then under the assumption of nesting, the intersection of the
172:
The
Bolzano–Weierstrass theorem is named after mathematicians
3643:
4599: – are precisely the closed and bounded subsets.
2752:
as the first interval for the sequence of nested intervals.
902:
comes after the final peak, which implies the existence of
3749:
has a convergent subsequence converging to an element of
769:
otherwise) and let the first index of a new subsequence
180:. It was actually first proved by Bolzano in 1817 as a
683:
has a monotone (non-increasing) subsequence, which is
4672:, then the system has a Pareto-efficient allocation.
4612:
4576:
4542:
4521:
4496:
4474:
4448:
4426:
4401:
4366:
4332:
4287:
4265:
4230:
4208:
4186:
4160:
4153:
is unbounded and therefore not convergent. Moreover,
4126:
4073:
4046:
4018:
3979:
3958:
3936:
3914:
3884:
3863:
3824:
3783:
3756:
3734:
3699:
3658:
3626:
3593:
3560:
3540:
3507:
3487:
3454:
3421:
3394:
3374:
3347:
3327:
3307:
3271:
3247:
3220:
3200:
3173:
3130:
3064:
3034:
2983:
2935:
2917:
again at the mid into two equally sized subintervals.
2896:
2857:
2809:
2770:
2713:
2681:
2661:
2613:
2568:
2527:
2494:
2441:
2397:
2361:
2334:
2298:
2242:
2196:
2160:
2104:
2071:
2035:
1991:
1955:
1922:
1878:
1842:
1755:
1726:
1693:
1673:
1644:
1611:
1565:
1545:
1519:
1490:
1461:
1428:
1399:
1370:
1326:
1290:
1257:
1177:
1123:
1083:
1056:
1029:
975:
935:
908:
881:
854:
815:
775:
749:
729:
689:
660:
627:
521:
443:
417:
377:
357:
307:
274:
239:
206:
125:
85:
52:
3415:
contains by construction infinitely many members of
2977:. We take this subinterval as the third subinterval
1829:{\displaystyle x_{n}=(x_{n1},x_{n2},\dots ,x_{nn})}
4798:(2nd ed.). Belmont, CA: Thomson Brooks/Cole.
4772:
4627:
4582:
4557:
4527:
4502:
4480:
4454:
4432:
4407:
4387:
4351:
4312:
4271:
4236:
4214:
4192:
4166:
4145:
4112:
4059:
4032:
4004:
3964:
3942:
3920:
3899:
3869:
3830:
3804:
3762:
3740:
3718:
3679:
3632:
3612:
3579:
3546:
3526:
3493:
3473:
3440:
3407:
3380:
3360:
3333:
3313:
3290:
3253:
3233:
3206:
3186:
3156:
3110:
3047:
2996:
2969:
2909:
2870:
2851:. We take this subinterval as the second interval
2843:
2783:
2744:
2687:
2667:
2647:
2587:
2542:
2513:
2480:
2427:
2383:
2347:
2320:
2284:
2228:
2182:
2146:
2090:
2057:
2021:
1977:
1941:
1908:
1864:
1828:
1741:
1712:
1679:
1659:
1630:
1597:
1551:
1531:
1505:
1476:
1447:
1414:
1385:
1341:
1305:
1276:
1243:
1163:
1109:
1069:
1042:
1015:
961:
921:
894:
867:
840:
801:
761:
735:
715:
675:
646:
613:
507:
429:
403:
363:
322:
293:
254:
221:
140:
100:
67:
4854:
4843:PlanetMath: proof of Bolzano–Weierstrass Theorem
4360:contains a subsequence converging to some point
2791:at the mid into two equally sized subintervals.
2229:{\displaystyle K_{2}\subseteq K_{1}\subseteq I}
1720:may be expressed as an n-tuple of sequences in
1422:. Firstly, we will acknowledge that a sequence
1251:, thereby proving that every infinite sequence
4767:
4569:has a subsequence converging to an element of
3929:has a subsequence converging to an element of
1050:comes after the final peak, hence there is an
351:: Let us call a positive integer-valued index
4346:
4333:
4301:
4288:
4140:
4127:
3993:
3980:
3713:
3700:
2655:is bounded, this sequence has a lower bound
371:of a sequence a "peak" of the sequence when
4793:
4646:
3805:{\displaystyle A\subseteq \mathbb {R} ^{n}}
3680:{\displaystyle A\subseteq \mathbb {R} ^{n}}
2970:{\displaystyle (x_{n})_{n\in \mathbb {N} }}
2844:{\displaystyle (x_{n})_{n\in \mathbb {N} }}
2648:{\displaystyle (x_{n})_{n\in \mathbb {N} }}
1559:is the index set of the sequence such that
167:
4838:A proof of the Bolzano–Weierstrass theorem
3644:Sequential compactness in Euclidean spaces
4615:
4545:
4375:
4026:
3907:with the property that every sequence in
3887:
3792:
3667:
3094:
2961:
2835:
2639:
2530:
1729:
1647:
1493:
1464:
1402:
1373:
1329:
1293:
663:
310:
242:
209:
128:
88:
55:
75:. The theorem states that each infinite
4223:converging to itself, must also lie in
2521:was arbitrary, any bounded sequence in
1164:{\displaystyle x_{n_{2}}\leq x_{n_{3}}}
4855:
3055:is a closed and bounded interval, say
1016:{\displaystyle x_{n_{1}}<x_{n_{2}}}
160:. The theorem is sometimes called the
115:. An equivalent formulation is that a
4715:Bartle and Sherbert 2000, p. 78 (for
4388:{\displaystyle x\in \mathbb {R} ^{n}}
3194:is not empty. Thus there is a number
2878:of the sequence of nested intervals.
2285:{\displaystyle (x_{n2})_{n\in K_{2}}}
2147:{\displaystyle (x_{n1})_{n\in K_{1}}}
743:be the final peak if one exists (let
4863:Theorems about real number sequences
4779:(3rd ed.). New York: J. Wiley.
4754:Bartle and Sherbert 2000, pp. 78-79.
4201:, which has a sequence of points in
4067:to be any arbitrary point such that
3501:contains infinitely many members of
3004:of the sequence of nested intervals.
2553:
2481:{\displaystyle (x_{n})_{n\in K_{n}}}
2029:. It follows then by the lemma that
4591:– i.e., the subsets that are
2562:. We start with a bounded sequence
13:
3587:. Thus, there is a subsequence of
14:
4879:
4813:
4606:, which asserts that a subset of
4033:{\displaystyle n\in \mathbb {N} }
1357:that this subsequence converges.
233:), in which case the ordering on
4794:Fitzpatrick, Patrick M. (2006).
4692:Completeness of the real numbers
4628:{\displaystyle \mathbb {R} ^{n}}
4558:{\displaystyle \mathbb {R} ^{n}}
3900:{\displaystyle \mathbb {R} ^{n}}
3474:{\displaystyle I_{N}\subseteq U}
3009:
2922:
2883:
2796:
2757:
2700:
2600:
2543:{\displaystyle \mathbb {R} ^{n}}
2091:{\displaystyle K_{1}\subseteq I}
1742:{\displaystyle \mathbb {R} ^{1}}
1660:{\displaystyle \mathbb {R} ^{n}}
1598:{\displaystyle (x_{n})_{n\in K}}
1506:{\displaystyle \mathbb {R} ^{n}}
1477:{\displaystyle \mathbb {R} ^{1}}
1415:{\displaystyle \mathbb {R} ^{1}}
1393:) can be reduced to the case of
1386:{\displaystyle \mathbb {R} ^{n}}
1342:{\displaystyle \mathbb {R} ^{1}}
1306:{\displaystyle \mathbb {R} ^{1}}
676:{\displaystyle \mathbb {R} ^{1}}
323:{\displaystyle \mathbb {R} ^{1}}
255:{\displaystyle \mathbb {R} ^{1}}
222:{\displaystyle \mathbb {R} ^{1}}
141:{\displaystyle \mathbb {R} ^{n}}
101:{\displaystyle \mathbb {R} ^{n}}
68:{\displaystyle \mathbb {R} ^{n}}
4697:Ekeland's variational principle
4659:allocation. An allocation is a
4113:{\displaystyle ||x_{n}||\geq n}
3157:{\displaystyle a_{n}\leq b_{n}}
404:{\displaystyle x_{m}\leq x_{n}}
200:First we prove the theorem for
4771:; Sherbert, Donald R. (2000).
4748:
4739:
4722:
4709:
4651:There are different important
4313:{\displaystyle \{x_{n}\}\in A}
4100:
4095:
4080:
4075:
4012:can be constructed. For every
4005:{\displaystyle \{x_{n}\}\in A}
3607:
3594:
3574:
3561:
3521:
3508:
3435:
3422:
3285:
3272:
3105:
3078:
2950:
2936:
2824:
2810:
2739:
2727:
2628:
2614:
2582:
2569:
2550:has a convergent subsequence.
2508:
2495:
2488:converges and therefore since
2456:
2442:
2428:{\displaystyle j=1,2,\dots ,n}
2378:
2362:
2315:
2299:
2260:
2243:
2177:
2161:
2122:
2105:
2052:
2036:
2022:{\displaystyle j=1,2,\dots ,n}
1972:
1956:
1936:
1923:
1909:{\displaystyle j=1,2,\dots ,n}
1859:
1843:
1823:
1769:
1707:
1694:
1625:
1612:
1580:
1566:
1442:
1429:
1271:
1258:
1110:{\displaystyle n_{2}<n_{3}}
962:{\displaystyle n_{1}<n_{2}}
796:
776:
710:
690:
641:
628:
337:(a subsequence that is either
288:
275:
162:sequential compactness theorem
1:
4821:"Bolzano-Weierstrass theorem"
4775:Introduction to Real Analysis
4761:
4728:Fitzpatrick 2006, p. 52 (for
4120:. Then, every subsequence of
4565:for which every sequence in
4251:(closed and bounded implies
3853:implies closed and bounded)
3554:is an accumulation point of
3028:, which states that if each
2154:converges. For the sequence
1667:and denote its index set by
1532:{\displaystyle K\subseteq I}
1355:monotone convergence theorem
1349:; by the lemma proven above
1313:has a monotone subsequence.
621:. So, the infinite sequence
7:
4848:The Bolzano-Weierstrass Rap
4826:Encyclopedia of Mathematics
4675:
4323:Bolzano-Weierstrass theorem
1638:be any bounded sequence in
802:{\displaystyle (x_{n_{j}})}
716:{\displaystyle (x_{n_{j}})}
29:Bolzano–Weierstrass theorem
10:
4884:
4682:Sequentially compact space
4321:is also bounded. From the
4175:must be closed, since any
268:: Every infinite sequence
186:intermediate value theorem
4745:Fitzpatrick 2006, p. xiv.
4352:{\displaystyle \{x_{n}\}}
4280:is bounded, any sequence
4146:{\displaystyle \{x_{n}\}}
3719:{\displaystyle \{x_{n}\}}
3214:that is in each interval
841:{\displaystyle n_{1}=N+1}
4702:
4647:Application to economics
2384:{\displaystyle (x_{nj})}
2321:{\displaystyle (x_{n2})}
2183:{\displaystyle (x_{n2})}
2058:{\displaystyle (x_{n1})}
1978:{\displaystyle (x_{nj})}
1865:{\displaystyle (x_{nj})}
515:and the following terms
195:
168:History and significance
43:in a finite-dimensional
41:result about convergence
3613:{\displaystyle (x_{n})}
3580:{\displaystyle (x_{n})}
3527:{\displaystyle (x_{n})}
3441:{\displaystyle (x_{n})}
3291:{\displaystyle (x_{n})}
2588:{\displaystyle (x_{n})}
2514:{\displaystyle (x_{n})}
1942:{\displaystyle (x_{n})}
1713:{\displaystyle (x_{n})}
1631:{\displaystyle (x_{n})}
1448:{\displaystyle (x_{n})}
1277:{\displaystyle (x_{n})}
647:{\displaystyle (x_{n})}
294:{\displaystyle (x_{n})}
4629:
4584:
4559:
4529:
4504:
4489:must be an element of
4482:
4456:
4434:
4409:
4389:
4353:
4314:
4273:
4253:sequential compactness
4238:
4216:
4194:
4168:
4147:
4114:
4061:
4034:
4006:
3966:
3944:
3922:
3901:
3871:
3851:sequential compactness
3832:
3806:
3764:
3742:
3720:
3681:
3634:
3614:
3581:
3548:
3528:
3495:
3475:
3442:
3409:
3382:
3362:
3335:
3315:
3292:
3255:
3235:
3208:
3188:
3158:
3112:
3111:{\displaystyle I_{n}=}
3049:
2998:
2971:
2911:
2872:
2845:
2785:
2746:
2745:{\displaystyle I_{1}=}
2689:
2669:
2649:
2589:
2544:
2515:
2482:
2429:
2385:
2349:
2322:
2286:
2230:
2184:
2148:
2092:
2059:
2023:
1979:
1943:
1910:
1866:
1830:
1743:
1714:
1681:
1661:
1632:
1599:
1553:
1533:
1507:
1478:
1449:
1416:
1387:
1343:
1316:Now suppose one has a
1307:
1278:
1245:
1165:
1111:
1071:
1044:
1017:
963:
923:
896:
869:
842:
803:
763:
737:
717:
677:
648:
615:
509:
431:
430:{\displaystyle m>n}
405:
365:
324:
295:
256:
223:
142:
102:
69:
4630:
4585:
4560:
4530:
4505:
4483:
4457:
4435:
4410:
4390:
4354:
4315:
4274:
4239:
4217:
4195:
4169:
4148:
4115:
4062:
4060:{\displaystyle x_{n}}
4035:
4007:
3967:
3945:
3923:
3902:
3872:
3833:
3807:
3765:
3743:
3721:
3682:
3635:
3615:
3582:
3549:
3529:
3496:
3476:
3443:
3410:
3408:{\displaystyle I_{N}}
3383:
3363:
3361:{\displaystyle I_{N}}
3336:
3316:
3301:Take a neighbourhood
3293:
3256:
3236:
3234:{\displaystyle I_{n}}
3209:
3189:
3187:{\displaystyle I_{n}}
3159:
3113:
3050:
3048:{\displaystyle I_{n}}
2999:
2997:{\displaystyle I_{3}}
2972:
2912:
2910:{\displaystyle I_{2}}
2873:
2871:{\displaystyle I_{2}}
2846:
2786:
2784:{\displaystyle I_{1}}
2747:
2690:
2670:
2650:
2590:
2545:
2516:
2483:
2430:
2386:
2350:
2348:{\displaystyle K_{n}}
2323:
2287:
2231:
2185:
2149:
2093:
2060:
2024:
1980:
1944:
1911:
1867:
1831:
1744:
1715:
1682:
1662:
1633:
1600:
1554:
1534:
1508:
1479:
1450:
1417:
1388:
1344:
1308:
1279:
1246:
1166:
1112:
1072:
1070:{\displaystyle n_{3}}
1045:
1043:{\displaystyle n_{2}}
1018:
964:
924:
922:{\displaystyle n_{2}}
897:
895:{\displaystyle n_{1}}
875:is not a peak, since
870:
868:{\displaystyle n_{1}}
843:
804:
764:
738:
718:
678:
649:
616:
510:
432:
406:
366:
325:
296:
257:
224:
152:if and only if it is
143:
103:
70:
4868:Compactness theorems
4610:
4593:sequentially compact
4574:
4540:
4519:
4494:
4472:
4446:
4424:
4399:
4364:
4330:
4285:
4263:
4228:
4206:
4184:
4158:
4124:
4071:
4044:
4016:
3977:
3956:
3934:
3912:
3882:
3861:
3822:
3815:sequentially compact
3781:
3754:
3732:
3697:
3690:sequentially compact
3656:
3624:
3591:
3558:
3538:
3505:
3485:
3452:
3419:
3392:
3372:
3368:that is a subset of
3345:
3325:
3305:
3269:
3245:
3241:. Now we show, that
3218:
3198:
3171:
3128:
3062:
3032:
2981:
2933:
2894:
2855:
2807:
2768:
2711:
2679:
2659:
2611:
2566:
2525:
2492:
2439:
2395:
2359:
2332:
2296:
2292:converges and hence
2240:
2194:
2158:
2102:
2069:
2033:
1989:
1985:is also bounded for
1953:
1920:
1876:
1840:
1753:
1724:
1691:
1671:
1642:
1609:
1563:
1543:
1517:
1488:
1459:
1426:
1397:
1368:
1324:
1288:
1255:
1175:
1121:
1081:
1054:
1027:
973:
933:
906:
879:
852:
813:
773:
747:
727:
687:
658:
625:
519:
441:
415:
375:
355:
305:
272:
237:
204:
184:in the proof of the
150:sequentially compact
123:
83:
50:
4687:Heine–Borel theorem
4666:preference relation
4604:Heine–Borel theorem
3534:. This proves that
2675:and an upper bound
762:{\displaystyle N=0}
39:, is a fundamental
4625:
4580:
4555:
4525:
4500:
4478:
4452:
4430:
4405:
4385:
4349:
4310:
4269:
4234:
4212:
4190:
4164:
4143:
4110:
4057:
4030:
4002:
3962:
3940:
3918:
3897:
3867:
3828:
3802:
3760:
3738:
3716:
3692:if every sequence
3677:
3630:
3620:that converges to
3610:
3577:
3544:
3524:
3491:
3471:
3438:
3405:
3378:
3358:
3331:
3311:
3288:
3263:accumulation point
3251:
3231:
3204:
3184:
3154:
3108:
3045:
2994:
2967:
2907:
2868:
2841:
2781:
2742:
2685:
2665:
2645:
2585:
2540:
2511:
2478:
2425:
2381:
2345:
2318:
2282:
2226:
2180:
2144:
2088:
2055:
2019:
1975:
1939:
1906:
1872:is a sequence for
1862:
1826:
1739:
1710:
1677:
1657:
1628:
1595:
1549:
1529:
1503:
1474:
1445:
1412:
1383:
1339:
1303:
1274:
1241:
1161:
1107:
1067:
1040:
1013:
959:
919:
892:
865:
838:
799:
759:
733:
713:
673:
644:
611:
505:
427:
401:
361:
320:
291:
252:
219:
138:
98:
65:
23:, specifically in
4796:Advanced Calculus
4769:Bartle, Robert G.
4597:subspace topology
4583:{\displaystyle A}
4528:{\displaystyle A}
4514:Thus the subsets
4503:{\displaystyle A}
4481:{\displaystyle x}
4455:{\displaystyle A}
4433:{\displaystyle A}
4408:{\displaystyle x}
4272:{\displaystyle A}
4237:{\displaystyle A}
4215:{\displaystyle A}
4193:{\displaystyle A}
4167:{\displaystyle A}
3965:{\displaystyle A}
3943:{\displaystyle A}
3921:{\displaystyle A}
3870:{\displaystyle A}
3831:{\displaystyle A}
3763:{\displaystyle A}
3741:{\displaystyle A}
3633:{\displaystyle x}
3547:{\displaystyle x}
3494:{\displaystyle U}
3381:{\displaystyle U}
3334:{\displaystyle x}
3314:{\displaystyle U}
3254:{\displaystyle x}
3207:{\displaystyle x}
2688:{\displaystyle S}
2668:{\displaystyle s}
2554:Alternative proof
1680:{\displaystyle I}
1552:{\displaystyle I}
736:{\displaystyle N}
364:{\displaystyle n}
4875:
4834:
4809:
4790:
4778:
4755:
4752:
4746:
4743:
4737:
4726:
4720:
4713:
4657:Pareto efficient
4641:metrizable space
4634:
4632:
4631:
4626:
4624:
4623:
4618:
4589:
4587:
4586:
4581:
4564:
4562:
4561:
4556:
4554:
4553:
4548:
4534:
4532:
4531:
4526:
4509:
4507:
4506:
4501:
4487:
4485:
4484:
4479:
4461:
4459:
4458:
4453:
4439:
4437:
4436:
4431:
4414:
4412:
4411:
4406:
4394:
4392:
4391:
4386:
4384:
4383:
4378:
4358:
4356:
4355:
4350:
4345:
4344:
4319:
4317:
4316:
4311:
4300:
4299:
4278:
4276:
4275:
4270:
4243:
4241:
4240:
4235:
4221:
4219:
4218:
4213:
4199:
4197:
4196:
4191:
4173:
4171:
4170:
4165:
4152:
4150:
4149:
4144:
4139:
4138:
4119:
4117:
4116:
4111:
4103:
4098:
4093:
4092:
4083:
4078:
4066:
4064:
4063:
4058:
4056:
4055:
4039:
4037:
4036:
4031:
4029:
4011:
4009:
4008:
4003:
3992:
3991:
3971:
3969:
3968:
3963:
3949:
3947:
3946:
3941:
3927:
3925:
3924:
3919:
3906:
3904:
3903:
3898:
3896:
3895:
3890:
3876:
3874:
3873:
3868:
3837:
3835:
3834:
3829:
3811:
3809:
3808:
3803:
3801:
3800:
3795:
3769:
3767:
3766:
3761:
3747:
3745:
3744:
3739:
3725:
3723:
3722:
3717:
3712:
3711:
3686:
3684:
3683:
3678:
3676:
3675:
3670:
3639:
3637:
3636:
3631:
3619:
3617:
3616:
3611:
3606:
3605:
3586:
3584:
3583:
3578:
3573:
3572:
3553:
3551:
3550:
3545:
3533:
3531:
3530:
3525:
3520:
3519:
3500:
3498:
3497:
3492:
3480:
3478:
3477:
3472:
3464:
3463:
3447:
3445:
3444:
3439:
3434:
3433:
3414:
3412:
3411:
3406:
3404:
3403:
3387:
3385:
3384:
3379:
3367:
3365:
3364:
3359:
3357:
3356:
3340:
3338:
3337:
3332:
3320:
3318:
3317:
3312:
3297:
3295:
3294:
3289:
3284:
3283:
3260:
3258:
3257:
3252:
3240:
3238:
3237:
3232:
3230:
3229:
3213:
3211:
3210:
3205:
3193:
3191:
3190:
3185:
3183:
3182:
3163:
3161:
3160:
3155:
3153:
3152:
3140:
3139:
3117:
3115:
3114:
3109:
3104:
3103:
3090:
3089:
3074:
3073:
3054:
3052:
3051:
3046:
3044:
3043:
3024:nested intervals
3013:
3003:
3001:
3000:
2995:
2993:
2992:
2976:
2974:
2973:
2968:
2966:
2965:
2964:
2948:
2947:
2926:
2916:
2914:
2913:
2908:
2906:
2905:
2887:
2877:
2875:
2874:
2869:
2867:
2866:
2850:
2848:
2847:
2842:
2840:
2839:
2838:
2822:
2821:
2800:
2790:
2788:
2787:
2782:
2780:
2779:
2761:
2751:
2749:
2748:
2743:
2723:
2722:
2704:
2694:
2692:
2691:
2686:
2674:
2672:
2671:
2666:
2654:
2652:
2651:
2646:
2644:
2643:
2642:
2626:
2625:
2604:
2594:
2592:
2591:
2586:
2581:
2580:
2560:nested intervals
2549:
2547:
2546:
2541:
2539:
2538:
2533:
2520:
2518:
2517:
2512:
2507:
2506:
2487:
2485:
2484:
2479:
2477:
2476:
2475:
2474:
2454:
2453:
2434:
2432:
2431:
2426:
2390:
2388:
2387:
2382:
2377:
2376:
2354:
2352:
2351:
2346:
2344:
2343:
2327:
2325:
2324:
2319:
2314:
2313:
2291:
2289:
2288:
2283:
2281:
2280:
2279:
2278:
2258:
2257:
2235:
2233:
2232:
2227:
2219:
2218:
2206:
2205:
2189:
2187:
2186:
2181:
2176:
2175:
2153:
2151:
2150:
2145:
2143:
2142:
2141:
2140:
2120:
2119:
2097:
2095:
2094:
2089:
2081:
2080:
2064:
2062:
2061:
2056:
2051:
2050:
2028:
2026:
2025:
2020:
1984:
1982:
1981:
1976:
1971:
1970:
1948:
1946:
1945:
1940:
1935:
1934:
1915:
1913:
1912:
1907:
1871:
1869:
1868:
1863:
1858:
1857:
1835:
1833:
1832:
1827:
1822:
1821:
1800:
1799:
1784:
1783:
1765:
1764:
1748:
1746:
1745:
1740:
1738:
1737:
1732:
1719:
1717:
1716:
1711:
1706:
1705:
1686:
1684:
1683:
1678:
1666:
1664:
1663:
1658:
1656:
1655:
1650:
1637:
1635:
1634:
1629:
1624:
1623:
1604:
1602:
1601:
1596:
1594:
1593:
1578:
1577:
1558:
1556:
1555:
1550:
1538:
1536:
1535:
1530:
1512:
1510:
1509:
1504:
1502:
1501:
1496:
1483:
1481:
1480:
1475:
1473:
1472:
1467:
1454:
1452:
1451:
1446:
1441:
1440:
1421:
1419:
1418:
1413:
1411:
1410:
1405:
1392:
1390:
1389:
1384:
1382:
1381:
1376:
1348:
1346:
1345:
1340:
1338:
1337:
1332:
1318:bounded sequence
1312:
1310:
1309:
1304:
1302:
1301:
1296:
1283:
1281:
1280:
1275:
1270:
1269:
1250:
1248:
1247:
1242:
1234:
1233:
1232:
1231:
1214:
1213:
1212:
1211:
1194:
1193:
1192:
1191:
1170:
1168:
1167:
1162:
1160:
1159:
1158:
1157:
1140:
1139:
1138:
1137:
1116:
1114:
1113:
1108:
1106:
1105:
1093:
1092:
1076:
1074:
1073:
1068:
1066:
1065:
1049:
1047:
1046:
1041:
1039:
1038:
1022:
1020:
1019:
1014:
1012:
1011:
1010:
1009:
992:
991:
990:
989:
968:
966:
965:
960:
958:
957:
945:
944:
928:
926:
925:
920:
918:
917:
901:
899:
898:
893:
891:
890:
874:
872:
871:
866:
864:
863:
847:
845:
844:
839:
825:
824:
808:
806:
805:
800:
795:
794:
793:
792:
768:
766:
765:
760:
742:
740:
739:
734:
722:
720:
719:
714:
709:
708:
707:
706:
682:
680:
679:
674:
672:
671:
666:
653:
651:
650:
645:
640:
639:
620:
618:
617:
612:
604:
603:
602:
601:
578:
577:
576:
575:
558:
557:
556:
555:
538:
537:
536:
535:
514:
512:
511:
506:
498:
497:
479:
478:
466:
465:
453:
452:
436:
434:
433:
428:
410:
408:
407:
402:
400:
399:
387:
386:
370:
368:
367:
362:
330:has an infinite
329:
327:
326:
321:
319:
318:
313:
300:
298:
297:
292:
287:
286:
261:
259:
258:
253:
251:
250:
245:
228:
226:
225:
220:
218:
217:
212:
178:Karl Weierstrass
147:
145:
144:
139:
137:
136:
131:
107:
105:
104:
99:
97:
96:
91:
77:bounded sequence
74:
72:
71:
66:
64:
63:
58:
37:Karl Weierstrass
4883:
4882:
4878:
4877:
4876:
4874:
4873:
4872:
4853:
4852:
4819:
4816:
4806:
4787:
4764:
4759:
4758:
4753:
4749:
4744:
4740:
4732:), p. 300 (for
4727:
4723:
4714:
4710:
4705:
4678:
4649:
4619:
4614:
4613:
4611:
4608:
4607:
4575:
4572:
4571:
4549:
4544:
4543:
4541:
4538:
4537:
4520:
4517:
4516:
4495:
4492:
4491:
4473:
4470:
4469:
4447:
4444:
4443:
4425:
4422:
4421:
4400:
4397:
4396:
4379:
4374:
4373:
4365:
4362:
4361:
4340:
4336:
4331:
4328:
4327:
4295:
4291:
4286:
4283:
4282:
4264:
4261:
4260:
4229:
4226:
4225:
4207:
4204:
4203:
4185:
4182:
4181:
4159:
4156:
4155:
4134:
4130:
4125:
4122:
4121:
4099:
4094:
4088:
4084:
4079:
4074:
4072:
4069:
4068:
4051:
4047:
4045:
4042:
4041:
4025:
4017:
4014:
4013:
3987:
3983:
3978:
3975:
3974:
3957:
3954:
3953:
3935:
3932:
3931:
3913:
3910:
3909:
3891:
3886:
3885:
3883:
3880:
3879:
3878:is a subset of
3862:
3859:
3858:
3823:
3820:
3819:
3817:if and only if
3796:
3791:
3790:
3782:
3779:
3778:
3755:
3752:
3751:
3733:
3730:
3729:
3707:
3703:
3698:
3695:
3694:
3671:
3666:
3665:
3657:
3654:
3653:
3646:
3625:
3622:
3621:
3601:
3597:
3592:
3589:
3588:
3568:
3564:
3559:
3556:
3555:
3539:
3536:
3535:
3515:
3511:
3506:
3503:
3502:
3486:
3483:
3482:
3459:
3455:
3453:
3450:
3449:
3429:
3425:
3420:
3417:
3416:
3399:
3395:
3393:
3390:
3389:
3373:
3370:
3369:
3352:
3348:
3346:
3343:
3342:
3326:
3323:
3322:
3306:
3303:
3302:
3279:
3275:
3270:
3267:
3266:
3246:
3243:
3242:
3225:
3221:
3219:
3216:
3215:
3199:
3196:
3195:
3178:
3174:
3172:
3169:
3168:
3148:
3144:
3135:
3131:
3129:
3126:
3125:
3099:
3095:
3085:
3081:
3069:
3065:
3063:
3060:
3059:
3039:
3035:
3033:
3030:
3029:
3017:
3014:
3005:
2988:
2984:
2982:
2979:
2978:
2960:
2953:
2949:
2943:
2939:
2934:
2931:
2930:
2927:
2918:
2901:
2897:
2895:
2892:
2891:
2888:
2879:
2862:
2858:
2856:
2853:
2852:
2834:
2827:
2823:
2817:
2813:
2808:
2805:
2804:
2801:
2792:
2775:
2771:
2769:
2766:
2765:
2762:
2753:
2718:
2714:
2712:
2709:
2708:
2705:
2696:
2680:
2677:
2676:
2660:
2657:
2656:
2638:
2631:
2627:
2621:
2617:
2612:
2609:
2608:
2605:
2576:
2572:
2567:
2564:
2563:
2556:
2534:
2529:
2528:
2526:
2523:
2522:
2502:
2498:
2493:
2490:
2489:
2470:
2466:
2459:
2455:
2449:
2445:
2440:
2437:
2436:
2396:
2393:
2392:
2369:
2365:
2360:
2357:
2356:
2339:
2335:
2333:
2330:
2329:
2306:
2302:
2297:
2294:
2293:
2274:
2270:
2263:
2259:
2250:
2246:
2241:
2238:
2237:
2214:
2210:
2201:
2197:
2195:
2192:
2191:
2168:
2164:
2159:
2156:
2155:
2136:
2132:
2125:
2121:
2112:
2108:
2103:
2100:
2099:
2076:
2072:
2070:
2067:
2066:
2043:
2039:
2034:
2031:
2030:
1990:
1987:
1986:
1963:
1959:
1954:
1951:
1950:
1930:
1926:
1921:
1918:
1917:
1877:
1874:
1873:
1850:
1846:
1841:
1838:
1837:
1814:
1810:
1792:
1788:
1776:
1772:
1760:
1756:
1754:
1751:
1750:
1733:
1728:
1727:
1725:
1722:
1721:
1701:
1697:
1692:
1689:
1688:
1687:. The sequence
1672:
1669:
1668:
1651:
1646:
1645:
1643:
1640:
1639:
1619:
1615:
1610:
1607:
1606:
1605:converges. Let
1583:
1579:
1573:
1569:
1564:
1561:
1560:
1544:
1541:
1540:
1518:
1515:
1514:
1497:
1492:
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1463:
1462:
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1457:
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1436:
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1406:
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1398:
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1133:
1129:
1128:
1124:
1122:
1119:
1118:
1101:
1097:
1088:
1084:
1082:
1079:
1078:
1061:
1057:
1055:
1052:
1051:
1034:
1030:
1028:
1025:
1024:
1005:
1001:
1000:
996:
985:
981:
980:
976:
974:
971:
970:
953:
949:
940:
936:
934:
931:
930:
913:
909:
907:
904:
903:
886:
882:
880:
877:
876:
859:
855:
853:
850:
849:
820:
816:
814:
811:
810:
788:
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783:
779:
774:
771:
770:
748:
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702:
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693:
688:
685:
684:
667:
662:
661:
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631:
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623:
622:
597:
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588:
571:
567:
566:
562:
551:
547:
546:
542:
531:
527:
526:
522:
520:
517:
516:
493:
489:
474:
470:
461:
457:
448:
444:
442:
439:
438:
416:
413:
412:
395:
391:
382:
378:
376:
373:
372:
356:
353:
352:
314:
309:
308:
306:
303:
302:
282:
278:
273:
270:
269:
246:
241:
240:
238:
235:
234:
213:
208:
207:
205:
202:
201:
198:
174:Bernard Bolzano
170:
132:
127:
126:
124:
121:
120:
92:
87:
86:
84:
81:
80:
59:
54:
53:
51:
48:
47:
45:Euclidean space
33:Bernard Bolzano
17:
12:
11:
5:
4881:
4871:
4870:
4865:
4851:
4850:
4845:
4840:
4835:
4815:
4814:External links
4812:
4811:
4810:
4804:
4791:
4785:
4763:
4760:
4757:
4756:
4747:
4738:
4721:
4707:
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4429:
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2959:
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2919:
2904:
2900:
2890:Then we split
2889:
2882:
2880:
2865:
2861:
2837:
2833:
2830:
2826:
2820:
2816:
2812:
2802:
2795:
2793:
2778:
2774:
2764:Then we split
2763:
2756:
2754:
2741:
2738:
2735:
2732:
2729:
2726:
2721:
2717:
2706:
2699:
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2448:
2444:
2424:
2421:
2418:
2415:
2412:
2409:
2406:
2403:
2400:
2391:converges for
2380:
2375:
2372:
2368:
2364:
2342:
2338:
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2309:
2305:
2301:
2277:
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2256:
2253:
2249:
2245:
2225:
2222:
2217:
2213:
2209:
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2200:
2179:
2174:
2171:
2167:
2163:
2139:
2135:
2131:
2128:
2124:
2118:
2115:
2111:
2107:
2087:
2084:
2079:
2075:
2054:
2049:
2046:
2042:
2038:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1974:
1969:
1966:
1962:
1958:
1938:
1933:
1929:
1925:
1905:
1902:
1899:
1896:
1893:
1890:
1887:
1884:
1881:
1861:
1856:
1853:
1849:
1845:
1825:
1820:
1817:
1813:
1809:
1806:
1803:
1798:
1795:
1791:
1787:
1782:
1779:
1775:
1771:
1768:
1763:
1759:
1736:
1731:
1709:
1704:
1700:
1696:
1676:
1654:
1649:
1627:
1622:
1618:
1614:
1592:
1589:
1586:
1582:
1576:
1572:
1568:
1548:
1528:
1525:
1522:
1500:
1495:
1471:
1466:
1444:
1439:
1435:
1431:
1409:
1404:
1380:
1375:
1336:
1331:
1300:
1295:
1273:
1268:
1264:
1260:
1240:
1237:
1230:
1226:
1221:
1217:
1210:
1206:
1201:
1197:
1190:
1186:
1181:
1156:
1152:
1147:
1143:
1136:
1132:
1127:
1104:
1100:
1096:
1091:
1087:
1064:
1060:
1037:
1033:
1008:
1004:
999:
995:
988:
984:
979:
956:
952:
948:
943:
939:
916:
912:
889:
885:
862:
858:
837:
834:
831:
828:
823:
819:
798:
791:
787:
782:
778:
758:
755:
752:
732:
712:
705:
701:
696:
692:
670:
665:
643:
638:
634:
630:
610:
607:
600:
596:
591:
587:
584:
581:
574:
570:
565:
561:
554:
550:
545:
541:
534:
530:
525:
504:
501:
496:
492:
488:
485:
482:
477:
473:
469:
464:
460:
456:
451:
447:
426:
423:
420:
398:
394:
390:
385:
381:
360:
343:non-increasing
339:non-decreasing
317:
312:
290:
285:
281:
277:
249:
244:
216:
211:
197:
194:
169:
166:
135:
130:
95:
90:
62:
57:
31:, named after
15:
9:
6:
4:
3:
2:
4880:
4869:
4866:
4864:
4861:
4860:
4858:
4849:
4846:
4844:
4841:
4839:
4836:
4832:
4828:
4827:
4822:
4818:
4817:
4807:
4805:0-534-37603-7
4801:
4797:
4792:
4788:
4786:9780471321484
4782:
4777:
4776:
4770:
4766:
4765:
4751:
4742:
4735:
4731:
4725:
4718:
4712:
4708:
4698:
4695:
4693:
4690:
4688:
4685:
4683:
4680:
4679:
4673:
4671:
4667:
4662:
4658:
4654:
4644:
4642:
4638:
4620:
4605:
4600:
4598:
4594:
4590:
4577:
4568:
4550:
4535:
4522:
4512:
4510:
4497:
4488:
4475:
4466:
4462:
4449:
4440:
4427:
4418:
4402:
4380:
4370:
4367:
4359:
4341:
4337:
4324:
4320:
4307:
4304:
4296:
4292:
4279:
4266:
4256:
4254:
4250:
4246:
4245:
4231:
4222:
4209:
4200:
4187:
4178:
4174:
4161:
4135:
4131:
4107:
4104:
4089:
4085:
4052:
4048:
4022:
4019:
3999:
3996:
3988:
3984:
3972:
3959:
3950:
3937:
3928:
3915:
3892:
3877:
3864:
3854:
3852:
3848:
3844:
3843:and bounded.
3842:
3838:
3825:
3816:
3812:
3797:
3787:
3784:
3776:
3772:
3770:
3757:
3748:
3735:
3726:
3708:
3704:
3691:
3687:
3672:
3662:
3659:
3650:
3641:
3627:
3602:
3598:
3569:
3565:
3541:
3516:
3512:
3488:
3468:
3465:
3460:
3456:
3430:
3426:
3400:
3396:
3375:
3353:
3349:
3328:
3308:
3299:
3280:
3276:
3264:
3248:
3226:
3222:
3201:
3179:
3175:
3149:
3145:
3141:
3136:
3132:
3124:
3123:
3122:
3100:
3096:
3091:
3086:
3082:
3075:
3070:
3066:
3058:
3057:
3056:
3040:
3036:
3027:
3025:
3012:
3007:
2989:
2985:
2957:
2954:
2944:
2940:
2925:
2920:
2902:
2898:
2886:
2881:
2863:
2859:
2831:
2828:
2818:
2814:
2799:
2794:
2776:
2772:
2760:
2755:
2736:
2733:
2730:
2724:
2719:
2715:
2703:
2698:
2682:
2662:
2635:
2632:
2622:
2618:
2603:
2598:
2597:
2596:
2577:
2573:
2561:
2551:
2535:
2503:
2499:
2471:
2467:
2463:
2460:
2450:
2446:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2373:
2370:
2366:
2340:
2336:
2310:
2307:
2303:
2275:
2271:
2267:
2264:
2254:
2251:
2247:
2223:
2220:
2215:
2211:
2207:
2202:
2198:
2172:
2169:
2165:
2137:
2133:
2129:
2126:
2116:
2113:
2109:
2085:
2082:
2077:
2073:
2047:
2044:
2040:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1967:
1964:
1960:
1931:
1927:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1854:
1851:
1847:
1818:
1815:
1811:
1807:
1804:
1801:
1796:
1793:
1789:
1785:
1780:
1777:
1773:
1766:
1761:
1757:
1734:
1702:
1698:
1674:
1652:
1620:
1616:
1590:
1587:
1584:
1574:
1570:
1546:
1526:
1523:
1520:
1498:
1469:
1437:
1433:
1407:
1378:
1363:
1358:
1356:
1352:
1334:
1319:
1314:
1298:
1266:
1262:
1238:
1235:
1228:
1224:
1219:
1215:
1208:
1204:
1199:
1195:
1188:
1184:
1179:
1154:
1150:
1145:
1141:
1134:
1130:
1125:
1102:
1098:
1094:
1089:
1085:
1062:
1058:
1035:
1031:
1006:
1002:
997:
993:
986:
982:
977:
954:
950:
946:
941:
937:
914:
910:
887:
883:
860:
856:
835:
832:
829:
826:
821:
817:
789:
785:
780:
756:
753:
750:
730:
703:
699:
694:
668:
636:
632:
608:
605:
598:
594:
589:
585:
582:
579:
572:
568:
563:
559:
552:
548:
543:
539:
532:
528:
523:
502:
499:
494:
490:
486:
483:
480:
475:
471:
467:
462:
458:
454:
449:
445:
424:
421:
418:
396:
392:
388:
383:
379:
358:
350:
346:
344:
340:
336:
333:
315:
283:
279:
267:
263:
247:
232:
214:
193:
191:
187:
183:
179:
175:
165:
163:
159:
155:
151:
133:
118:
114:
111:
93:
78:
60:
46:
42:
38:
34:
30:
26:
25:real analysis
22:
4824:
4795:
4774:
4750:
4741:
4733:
4729:
4724:
4716:
4711:
4650:
4601:
4570:
4566:
4515:
4513:
4490:
4468:
4442:
4420:
4326:
4281:
4259:
4257:
4248:
4247:
4224:
4202:
4180:
4154:
3952:
3930:
3908:
3857:
3855:
3846:
3845:
3818:
3777:
3774:
3773:
3750:
3728:
3693:
3652:
3648:
3647:
3300:
3166:
3120:
3022:
3020:
2557:
1949:is bounded,
1362:general case
1359:
1351:there exists
1315:
348:
347:
265:
264:
231:real numbers
229:(set of all
199:
171:
161:
28:
18:
4653:equilibrium
4417:limit point
4177:limit point
3649:Definition:
335:subsequence
113:subsequence
21:mathematics
4857:Categories
4762:References
4465:closed set
3388:. Because
2355:for which
2236:such that
2098:such that
1749:such that
809:be set to
411:for every
110:convergent
4831:EMS Press
4670:non-empty
4371:∈
4305:∈
4105:≥
4040:, define
4023:∈
3997:∈
3788:⊆
3663:⊆
3466:⊆
3142:≤
2958:∈
2832:∈
2636:∈
2464:∈
2435:. Hence,
2417:…
2268:∈
2221:⊆
2208:⊆
2130:∈
2083:⊆
2011:…
1898:…
1805:…
1588:∈
1524:⊆
1239:…
1236:≤
1216:≤
1196:≤
1142:≤
1023:. Again,
609:…
606:≥
586:≥
583:⋯
580:≥
560:≥
540:≥
503:…
484:⋯
389:≤
4676:See also
4395:. Since
3856:Suppose
3775:Theorem:
2707:We take
2607:Because
1916:. Since
848:. Then
332:monotone
190:analysis
4833:, 2001
4637:compact
4595:in the
3951:. Then
3481:, also
3026:theorem
158:bounded
4802:
4783:
4661:matrix
4258:Since
4249:Proof:
3847:Proof:
3841:closed
3651:A set
3261:is an
1836:where
1539:where
1077:where
154:closed
117:subset
108:has a
27:, the
4703:Notes
4463:is a
4415:is a
3121:with
1117:with
929:with
349:Proof
266:Lemma
196:Proof
182:lemma
4800:ISBN
4781:ISBN
4441:and
3448:and
1455:(in
1360:The
1095:<
994:<
969:and
947:<
500:<
487:<
481:<
468:<
455:<
422:>
176:and
156:and
35:and
4635:is
4536:of
4419:of
4179:of
3839:is
3813:is
3727:in
3688:is
3321:of
3265:of
1484:or
1320:in
1284:in
654:in
345:).
341:or
301:in
148:is
119:of
79:in
19:In
4859::
4829:,
4823:,
4736:).
4719:).
4511:.
4467:,
4325:,
4255:)
3771:.
3640:.
3298:.
2595::
192:.
164:.
4808:.
4789:.
4734:R
4730:R
4717:R
4621:n
4616:R
4578:A
4567:A
4551:n
4546:R
4523:A
4498:A
4476:x
4450:A
4428:A
4403:x
4381:n
4376:R
4368:x
4347:}
4342:n
4338:x
4334:{
4308:A
4302:}
4297:n
4293:x
4289:{
4267:A
4244:.
4232:A
4210:A
4188:A
4162:A
4141:}
4136:n
4132:x
4128:{
4108:n
4101:|
4096:|
4090:n
4086:x
4081:|
4076:|
4053:n
4049:x
4027:N
4020:n
4000:A
3994:}
3989:n
3985:x
3981:{
3960:A
3938:A
3916:A
3893:n
3888:R
3865:A
3849:(
3826:A
3798:n
3793:R
3785:A
3758:A
3736:A
3714:}
3709:n
3705:x
3701:{
3673:n
3668:R
3660:A
3628:x
3608:)
3603:n
3599:x
3595:(
3575:)
3570:n
3566:x
3562:(
3542:x
3522:)
3517:n
3513:x
3509:(
3489:U
3469:U
3461:N
3457:I
3436:)
3431:n
3427:x
3423:(
3401:N
3397:I
3376:U
3354:N
3350:I
3329:x
3309:U
3286:)
3281:n
3277:x
3273:(
3249:x
3227:n
3223:I
3202:x
3180:n
3176:I
3150:n
3146:b
3137:n
3133:a
3106:]
3101:n
3097:b
3092:,
3087:n
3083:a
3079:[
3076:=
3071:n
3067:I
3041:n
3037:I
2990:3
2986:I
2962:N
2955:n
2951:)
2945:n
2941:x
2937:(
2903:2
2899:I
2864:2
2860:I
2836:N
2829:n
2825:)
2819:n
2815:x
2811:(
2777:1
2773:I
2740:]
2737:S
2734:,
2731:s
2728:[
2725:=
2720:1
2716:I
2695:.
2683:S
2663:s
2640:N
2633:n
2629:)
2623:n
2619:x
2615:(
2583:)
2578:n
2574:x
2570:(
2536:n
2531:R
2509:)
2504:n
2500:x
2496:(
2472:n
2468:K
2461:n
2457:)
2451:n
2447:x
2443:(
2423:n
2420:,
2414:,
2411:2
2408:,
2405:1
2402:=
2399:j
2379:)
2374:j
2371:n
2367:x
2363:(
2341:n
2337:K
2316:)
2311:2
2308:n
2304:x
2300:(
2276:2
2272:K
2265:n
2261:)
2255:2
2252:n
2248:x
2244:(
2224:I
2216:1
2212:K
2203:2
2199:K
2178:)
2173:2
2170:n
2166:x
2162:(
2138:1
2134:K
2127:n
2123:)
2117:1
2114:n
2110:x
2106:(
2086:I
2078:1
2074:K
2053:)
2048:1
2045:n
2041:x
2037:(
2017:n
2014:,
2008:,
2005:2
2002:,
1999:1
1996:=
1993:j
1973:)
1968:j
1965:n
1961:x
1957:(
1937:)
1932:n
1928:x
1924:(
1904:n
1901:,
1895:,
1892:2
1889:,
1886:1
1883:=
1880:j
1860:)
1855:j
1852:n
1848:x
1844:(
1824:)
1819:n
1816:n
1812:x
1808:,
1802:,
1797:2
1794:n
1790:x
1786:,
1781:1
1778:n
1774:x
1770:(
1767:=
1762:n
1758:x
1735:1
1730:R
1708:)
1703:n
1699:x
1695:(
1675:I
1653:n
1648:R
1626:)
1621:n
1617:x
1613:(
1591:K
1585:n
1581:)
1575:n
1571:x
1567:(
1547:I
1527:I
1521:K
1499:n
1494:R
1470:1
1465:R
1443:)
1438:n
1434:x
1430:(
1408:1
1403:R
1379:n
1374:R
1364:(
1335:1
1330:R
1299:1
1294:R
1272:)
1267:n
1263:x
1259:(
1229:3
1225:n
1220:x
1209:2
1205:n
1200:x
1189:1
1185:n
1180:x
1155:3
1151:n
1146:x
1135:2
1131:n
1126:x
1103:3
1099:n
1090:2
1086:n
1063:3
1059:n
1036:2
1032:n
1007:2
1003:n
998:x
987:1
983:n
978:x
955:2
951:n
942:1
938:n
915:2
911:n
888:1
884:n
861:1
857:n
836:1
833:+
830:N
827:=
822:1
818:n
797:)
790:j
786:n
781:x
777:(
757:0
754:=
751:N
731:N
711:)
704:j
700:n
695:x
691:(
669:1
664:R
642:)
637:n
633:x
629:(
599:j
595:n
590:x
573:3
569:n
564:x
553:2
549:n
544:x
533:1
529:n
524:x
495:j
491:n
476:3
472:n
463:2
459:n
450:1
446:n
425:n
419:m
397:n
393:x
384:m
380:x
359:n
316:1
311:R
289:)
284:n
280:x
276:(
248:1
243:R
215:1
210:R
134:n
129:R
94:n
89:R
61:n
56:R
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