1231:
states that every continuous function that attains both negative and positive values has a root. This is a consequence of the least upper bound property, but it can also be used to prove the least upper bound property if treated as an axiom. (The definition of continuity does not depend on any form
1179:
where the law of the excluded middle does not hold, the full form of the least upper bound property fails for the
Dedekind reals, while the open induction property remains true in most models (following from Brouwer's bar theorem) and is strong enough to give short proofs of key theorems.
112:. Different versions of this axiom are all equivalent in the sense that any ordered field that satisfies one form of completeness satisfies all of them, apart from Cauchy completeness and nested intervals theorem, which are strictly weaker in that there are
1198:
by Körner) states that every nondecreasing, bounded sequence of real numbers converges. This can be viewed as a special case of the least upper bound property, but it can also be used fairly directly to prove the Cauchy completeness of the real numbers.
998:
Nested intervals theorem shares the same logical status as Cauchy completeness in this spectrum of expressions of completeness. In other words, nested intervals theorem by itself is weaker than other forms of completeness, although taken together with
603:. If one were to repeat the construction of real numbers with Dedekind cuts (i.e., "close" the set of real numbers by adding all possible Dedekind cuts), one would obtain no additional numbers because the real numbers are already Dedekind complete.
688:
832:
986:
556:
481:
1175:
The open induction principle can be shown to be equivalent to
Dedekind completeness for arbitrary ordered sets under the order topology, using proofs by contradiction. In weaker foundations such as in
708:
Cauchy completeness is related to the construction of the real numbers using Cauchy sequences. Essentially, this method defines a real number to be the limit of a Cauchy sequence of rational numbers.
226:
991:
is a nested sequence of closed intervals in the rational numbers whose intersection is empty. (In the real numbers, the intersection of these intervals contains the number
396:
of the real numbers is generated by a real number. In a synthetic approach to the real numbers, this is the version of completeness that is most often included as an axiom.
601:
1170:
1137:
1099:
1516:
705:. Though this is a Cauchy sequence of rational numbers, it does not converge to any rational number. (In this real number line, this sequence converges to pi.)
1232:
of completeness, so there is no circularity: what is meant is that the intermediate value theorem and the least upper bound property are equivalent statements.)
1029:
1061:
1190:
638:
572:
772:
891:
487:
412:
101:
1561:
1466:
1435:
169:
119:
that are ordered and Cauchy complete. When the real numbers are instead constructed using a model, completeness becomes a
1427:
1320:
1412:
1393:
1370:
1351:
1328:
1305:
1273:
881:
does not satisfy the nested interval theorem. For example, the sequence (whose terms are derived from the digits of
1511:
730:, Cauchy completeness is weaker than the other forms of completeness on this page. But Cauchy completeness and the
57:
1209:
113:
1241:
33:
that, intuitively, implies that there are no "gaps" (in
Dedekind's terminology) or "missing points" in the
370:
1459:
1227:
132:
49:, completeness is equivalent to the statement that any infinite string of decimal digits is actually a
1521:
1501:
1293:
862:
1104:
1261:
582:
575:
based on the idea of using
Dedekind cuts of rational numbers to name real numbers; e.g. the cut
763:
50:
1587:
1551:
1452:
1176:
720:
712:
384:
366:
85:
163:
does not have the least upper bound property. An example is the subset of rational numbers
1546:
1066:
1000:
731:
619:
116:
20:
8:
1491:
73:
1340:
1014:
1034:
1531:
1431:
1408:
1389:
1382:
1366:
1347:
1324:
1301:
1269:
1217:. Again, this theorem is equivalent to the other forms of completeness given above.
150:
42:
1388:. Walter Rudin Student Series in Advanced Mathematics (3rd ed.). McGraw-Hill.
743:
632:
is not Cauchy complete. An example is the following sequence of rational numbers:
1541:
878:
626:
615:
400:
157:
38:
1526:
1581:
1536:
727:
715:, Cauchy completeness can be generalized to a notion of completeness for any
683:{\displaystyle 3,\quad 3.1,\quad 3.14,\quad 3.142,\quad 3.1416,\quad \ldots }
105:
568:
does not have a minimum, so this cut is not generated by a rational number.
231:
This set has an upper bound. However, this set has no least upper bound in
1266:
A companion to analysis: a second first and first second course in analysis
716:
393:
30:
827:{\displaystyle I_{1}\;\supset \;I_{2}\;\supset \;I_{3}\;\supset \;\cdots }
1475:
1214:
146:
97:
1566:
1496:
981:{\displaystyle \;\supset \;\;\supset \;\;\supset \;\;\supset \;\cdots }
1556:
551:{\displaystyle R=\{x\in \mathbb {Q} \mid x^{2}\geq 2\wedge x>0\}.}
34:
1337:
1213:
states that every bounded sequence of real numbers has a convergent
365:
The least upper bound property can be generalized to the setting of
1486:
476:{\displaystyle L=\{x\in \mathbb {Q} \mid x^{2}\leq 2\vee x<0\}.}
142:
1365:. Undergraduate Texts in Mathematics. New York: Springer Verlag.
1011:
The open induction principle states that a non-empty open subset
120:
69:
46:
1402:
766:, and suppose that these intervals are nested in the sense that
1444:
61:
237:: the least upper bound as a subset of the reals would be
406:
is not
Dedekind complete. An example is the Dedekind cut
992:
882:
702:
352:, etc., such that we never find a least-upper-bound of
221:{\displaystyle S=\{x\in \mathbb {Q} \mid x^{2}<2\}.}
41:, whose corresponding number line has a "gap" at each
1140:
1107:
1069:
1037:
1017:
894:
775:
641:
585:
490:
415:
337:. We can proceed similarly to find an upper bound of
172:
1346:(3rd ed.). New York City: John Wiley and Sons.
1292:
314:. However, we can choose a smaller upper bound, say
1381:
1339:
1164:
1131:
1093:
1055:
1023:
980:
826:
682:
595:
550:
475:
220:
1220:
1063:must be equal to the entire interval, if for any
392:Dedekind completeness is the property that every
1579:
76:forms of completeness, the most prominent being
1421:
1360:
1338:Bartle, Robert G.; Sherbert, Donald R. (2000).
1314:
1183:
1006:
1202:
861:. The nested interval theorem states that the
72:proven from the construction. There are many
1460:
1379:
734:taken together are equivalent to the others.
325:for the same reasons, but it is smaller than
126:
542:
497:
467:
422:
387:for more general concepts bearing this name.
212:
179:
1403:Dangello, Frank; Seyfried, Michael (1999).
737:
60:used, completeness may take the form of an
1467:
1453:
974:
970:
954:
950:
934:
930:
914:
910:
820:
816:
805:
801:
790:
786:
153:(or supremum) in the set of real numbers.
1254:
507:
432:
189:
376:
91:
1580:
1317:Mathematical Analysis: An Introduction
1260:
752:is another form of completeness. Let
606:
1448:
1517:Decidability of first-order theories
1424:A Radical Approach to Real Analysis
1384:Principles of Mathematical Analysis
13:
1323:. New York City: Springer Verlag.
1321:Undergraduate Texts in Mathematics
1286:
1003:, it is equivalent to the others.
14:
1599:
321:; this is also an upper bound of
145:subset of real numbers having an
1474:
573:construction of the real numbers
58:construction of the real numbers
697:th term in the sequence is the
676:
669:
662:
655:
648:
291:is certainly an upper bound of
257:, there is another upper bound
149:(or bounded above) must have a
108:satisfying some version of the
1221:The intermediate value theorem
1153:
1141:
1132:{\displaystyle [a,r)\subset S}
1120:
1108:
1088:
1076:
1050:
1038:
967:
955:
947:
935:
927:
915:
907:
895:
333:is not a least-upper-bound of
86:completeness as a metric space
1:
1342:Introduction to Real Analysis
1247:
1196:fundamental axiom of analysis
701:th decimal approximation for
1242:List of real analysis topics
1191:monotone convergence theorem
1184:Monotone convergence theorem
1007:The open induction principle
874:contains exactly one point.
614:is the statement that every
564:does not have a maximum and
7:
1298:Principles of real analysis
1296:; Burkinshaw, Owen (1998).
1235:
1210:Bolzano–Weierstrass theorem
1203:Bolzano–Weierstrass theorem
596:{\displaystyle {\sqrt {2}}}
579:described above would name
371:completeness (order theory)
241:, but it does not exist in
123:or collection of theorems.
37:. This contrasts with the
10:
1604:
1405:Introductory Real Analysis
1300:(3rd ed.). Academic.
1294:Aliprantis, Charalambos D.
1228:intermediate value theorem
741:
139:least-upper-bound property
133:Least-upper-bound property
130:
127:Least upper bound property
18:
1522:Extended real number line
1482:
1165:{\displaystyle \subset S}
306:; that is, no element of
1422:Bressoud, David (2007).
1361:Abbott, Stephen (2001).
1315:Browder, Andrew (1996).
865:of all of the intervals
762:be a sequence of closed
738:Nested intervals theorem
19:Not to be confused with
750:nested interval theorem
1380:Rudin, Walter (1976).
1363:Understanding Analysis
1262:Körner, Thomas William
1166:
1133:
1095:
1057:
1025:
982:
885:in the suggested way)
837:Moreover, assume that
828:
684:
597:
552:
477:
367:partially ordered sets
247:. For any upper bound
222:
53:for some real number.
51:decimal representation
16:Concept in mathematics
1562:Tarski axiomatization
1552:Real coordinate space
1502:Cantor–Dedekind axiom
1177:constructive analysis
1167:
1134:
1096:
1094:{\displaystyle r\in }
1058:
1026:
983:
829:
721:complete metric space
713:mathematical analysis
685:
598:
553:
478:
385:Dedekind completeness
377:Dedekind completeness
341:that is smaller than
223:
102:defined synthetically
92:Forms of completeness
78:Dedekind completeness
47:decimal number system
29:is a property of the
1547:Rational zeta series
1138:
1105:
1067:
1035:
1015:
1001:Archimedean property
892:
879:rational number line
773:
732:Archimedean property
639:
627:rational number line
583:
488:
413:
401:rational number line
170:
158:rational number line
21:Completeness (logic)
1492:Absolute difference
612:Cauchy completeness
607:Cauchy completeness
280:For instance, take
82:Cauchy completeness
1194:(described as the
1162:
1129:
1091:
1053:
1021:
978:
824:
680:
622:to a real number.
593:
548:
473:
218:
141:states that every
117:Archimedean fields
110:completeness axiom
66:completeness axiom
1575:
1574:
1532:Irrational number
1437:978-0-88385-747-2
1024:{\displaystyle S}
591:
151:least upper bound
56:Depending on the
1595:
1469:
1462:
1455:
1446:
1445:
1441:
1418:
1399:
1387:
1376:
1357:
1345:
1334:
1311:
1280:
1279:
1258:
1171:
1169:
1168:
1163:
1136:
1135:
1130:
1100:
1098:
1097:
1092:
1062:
1060:
1059:
1056:{\displaystyle }
1054:
1031:of the interval
1030:
1028:
1027:
1022:
987:
985:
984:
979:
873:
860:
853:
833:
831:
830:
825:
815:
814:
800:
799:
785:
784:
761:
744:Nested intervals
689:
687:
686:
681:
618:of real numbers
602:
600:
599:
594:
592:
587:
557:
555:
554:
549:
523:
522:
510:
482:
480:
479:
474:
448:
447:
435:
361:
355:
351:
344:
340:
336:
332:
328:
324:
320:
313:
309:
305:
299:is positive and
298:
294:
290:
286:
276:
266:
256:
246:
240:
236:
227:
225:
224:
219:
205:
204:
192:
39:rational numbers
35:real number line
1603:
1602:
1598:
1597:
1596:
1594:
1593:
1592:
1578:
1577:
1576:
1571:
1542:Rational number
1478:
1473:
1438:
1415:
1407:. Brooks Cole.
1396:
1373:
1354:
1331:
1308:
1289:
1287:Further reading
1284:
1283:
1276:
1268:. AMS Chelsea.
1259:
1255:
1250:
1238:
1223:
1205:
1186:
1139:
1106:
1103:
1102:
1101:, we have that
1068:
1065:
1064:
1036:
1033:
1032:
1016:
1013:
1012:
1009:
893:
890:
889:
871:
866:
855:
850:
843:
838:
810:
806:
795:
791:
780:
776:
774:
771:
770:
758:
753:
746:
740:
640:
637:
636:
616:Cauchy sequence
609:
586:
584:
581:
580:
518:
514:
506:
489:
486:
485:
443:
439:
431:
414:
411:
410:
379:
357:
353:
346:
342:
338:
334:
330:
326:
322:
315:
311:
310:is larger than
307:
300:
296:
292:
288:
281:
268:
258:
248:
242:
238:
232:
200:
196:
188:
171:
168:
167:
135:
129:
94:
68:), or may be a
45:value. In the
24:
17:
12:
11:
5:
1601:
1591:
1590:
1573:
1572:
1570:
1569:
1564:
1559:
1554:
1549:
1544:
1539:
1534:
1529:
1527:Gregory number
1524:
1519:
1514:
1509:
1504:
1499:
1494:
1489:
1483:
1480:
1479:
1472:
1471:
1464:
1457:
1449:
1443:
1442:
1436:
1419:
1413:
1400:
1394:
1377:
1371:
1358:
1352:
1335:
1329:
1312:
1306:
1288:
1285:
1282:
1281:
1274:
1252:
1251:
1249:
1246:
1245:
1244:
1237:
1234:
1222:
1219:
1204:
1201:
1185:
1182:
1161:
1158:
1155:
1152:
1149:
1146:
1143:
1128:
1125:
1122:
1119:
1116:
1113:
1110:
1090:
1087:
1084:
1081:
1078:
1075:
1072:
1052:
1049:
1046:
1043:
1040:
1020:
1008:
1005:
989:
988:
977:
973:
969:
966:
963:
960:
957:
953:
949:
946:
943:
940:
937:
933:
929:
926:
923:
920:
917:
913:
909:
906:
903:
900:
897:
869:
848:
841:
835:
834:
823:
819:
813:
809:
804:
798:
794:
789:
783:
779:
756:
742:Main article:
739:
736:
691:
690:
679:
675:
672:
668:
665:
661:
658:
654:
651:
647:
644:
608:
605:
590:
559:
558:
547:
544:
541:
538:
535:
532:
529:
526:
521:
517:
513:
509:
505:
502:
499:
496:
493:
483:
472:
469:
466:
463:
460:
457:
454:
451:
446:
442:
438:
434:
430:
427:
424:
421:
418:
390:
389:
378:
375:
229:
228:
217:
214:
211:
208:
203:
199:
195:
191:
187:
184:
181:
178:
175:
131:Main article:
128:
125:
93:
90:
15:
9:
6:
4:
3:
2:
1600:
1589:
1586:
1585:
1583:
1568:
1565:
1563:
1560:
1558:
1555:
1553:
1550:
1548:
1545:
1543:
1540:
1538:
1537:Normal number
1535:
1533:
1530:
1528:
1525:
1523:
1520:
1518:
1515:
1513:
1510:
1508:
1505:
1503:
1500:
1498:
1495:
1493:
1490:
1488:
1485:
1484:
1481:
1477:
1470:
1465:
1463:
1458:
1456:
1451:
1450:
1447:
1439:
1433:
1429:
1425:
1420:
1416:
1414:9780395959336
1410:
1406:
1401:
1397:
1395:9780070542358
1391:
1386:
1385:
1378:
1374:
1372:0-387-95060-5
1368:
1364:
1359:
1355:
1353:0-471-32148-6
1349:
1344:
1343:
1336:
1332:
1330:0-387-94614-4
1326:
1322:
1318:
1313:
1309:
1307:0-12-050257-7
1303:
1299:
1295:
1291:
1290:
1277:
1275:9780821834473
1271:
1267:
1263:
1257:
1253:
1243:
1240:
1239:
1233:
1230:
1229:
1218:
1216:
1212:
1211:
1200:
1197:
1193:
1192:
1181:
1178:
1173:
1159:
1156:
1150:
1147:
1144:
1126:
1123:
1117:
1114:
1111:
1085:
1082:
1079:
1073:
1070:
1047:
1044:
1041:
1018:
1004:
1002:
996:
994:
975:
971:
964:
961:
958:
951:
944:
941:
938:
931:
924:
921:
918:
911:
904:
901:
898:
888:
887:
886:
884:
880:
875:
872:
864:
858:
851:
844:
821:
817:
811:
807:
802:
796:
792:
787:
781:
777:
769:
768:
767:
765:
759:
751:
745:
735:
733:
729:
728:ordered field
724:
722:
718:
714:
709:
706:
704:
700:
696:
677:
673:
670:
666:
663:
659:
656:
652:
649:
645:
642:
635:
634:
633:
631:
628:
623:
621:
617:
613:
604:
588:
578:
574:
569:
567:
563:
545:
539:
536:
533:
530:
527:
524:
519:
515:
511:
503:
500:
494:
491:
484:
470:
464:
461:
458:
455:
452:
449:
444:
440:
436:
428:
425:
419:
416:
409:
408:
407:
405:
402:
397:
395:
388:
386:
381:
380:
374:
372:
368:
363:
360:
349:
318:
303:
284:
278:
275:
271:
265:
261:
255:
251:
245:
235:
215:
209:
206:
201:
197:
193:
185:
182:
176:
173:
166:
165:
164:
162:
159:
154:
152:
148:
144:
140:
134:
124:
122:
118:
115:
111:
107:
106:ordered field
103:
99:
89:
87:
83:
79:
75:
71:
67:
63:
59:
54:
52:
48:
44:
40:
36:
32:
28:
22:
1588:Real numbers
1512:Construction
1507:Completeness
1506:
1476:Real numbers
1423:
1404:
1383:
1362:
1341:
1316:
1297:
1265:
1256:
1226:
1224:
1208:
1206:
1195:
1189:
1187:
1174:
1010:
997:
990:
876:
867:
863:intersection
856:
846:
839:
836:
754:
749:
747:
725:
717:metric space
710:
707:
698:
694:
692:
629:
624:
611:
610:
576:
570:
565:
561:
560:
403:
398:
394:Dedekind cut
391:
382:
364:
358:
347:
316:
301:
282:
279:
273:
269:
263:
259:
253:
249:
243:
233:
230:
160:
155:
138:
136:
109:
98:real numbers
95:
81:
77:
65:
55:
31:real numbers
27:Completeness
26:
25:
1215:subsequence
571:There is a
147:upper bound
1567:Vitali set
1497:Cantor set
1248:References
304:= 2.25 ≥ 2
74:equivalent
43:irrational
1557:Real line
1157:⊂
1124:⊂
1074:∈
976:⋯
972:⊃
952:⊃
932:⊃
912:⊃
822:⋯
818:⊃
803:⊃
788:⊃
764:intervals
693:Here the
678:…
620:converges
531:∧
525:≥
512:∣
504:∈
456:∨
450:≤
437:∣
429:∈
194:∣
186:∈
1582:Category
1487:0.999...
1264:(2004).
1236:See also
295:, since
143:nonempty
726:For an
719:. See
369:. See
287:, then
121:theorem
100:can be
70:theorem
1434:
1411:
1392:
1369:
1350:
1327:
1304:
1272:
671:3.1416
350:= 1.42
345:, say
319:= 1.45
104:as an
965:3.142
959:3.141
664:3.142
577:(L,R)
329:, so
285:= 1.5
272:<
267:with
64:(the
62:axiom
1432:ISBN
1409:ISBN
1390:ISBN
1367:ISBN
1348:ISBN
1325:ISBN
1302:ISBN
1270:ISBN
1225:The
1207:The
1188:The
945:3.15
939:3.14
877:The
859:→ +∞
748:The
657:3.14
625:The
537:>
462:<
399:The
383:See
207:<
156:The
137:The
96:The
80:and
1428:MAA
995:.)
925:3.2
919:3.1
854:as
852:→ 0
711:In
650:3.1
356:in
114:non
88:).
1584::
1430:.
1426:.
1319:.
1172:.
993:pi
883:pi
845:−
760:=
723:.
703:pi
373:.
362:.
277:.
262:∈
252:∈
239:√2
1468:e
1461:t
1454:v
1440:.
1417:.
1398:.
1375:.
1356:.
1333:.
1310:.
1278:.
1160:S
1154:]
1151:r
1148:,
1145:a
1142:[
1127:S
1121:)
1118:r
1115:,
1112:a
1109:[
1089:]
1086:b
1083:,
1080:a
1077:[
1071:r
1051:]
1048:b
1045:,
1042:a
1039:[
1019:S
968:]
962:,
956:[
948:]
942:,
936:[
928:]
922:,
916:[
908:]
905:4
902:,
899:3
896:[
870:n
868:I
857:n
849:n
847:a
842:n
840:b
812:3
808:I
797:2
793:I
782:1
778:I
757:n
755:I
699:n
695:n
674:,
667:,
660:,
653:,
646:,
643:3
630:Q
589:2
566:R
562:L
546:.
543:}
540:0
534:x
528:2
520:2
516:x
508:Q
501:x
498:{
495:=
492:R
471:.
468:}
465:0
459:x
453:2
445:2
441:x
433:Q
426:x
423:{
420:=
417:L
404:Q
359:Q
354:S
348:z
343:y
339:S
335:S
331:x
327:x
323:S
317:y
312:x
308:S
302:x
297:x
293:S
289:x
283:x
274:x
270:y
264:Q
260:y
254:Q
250:x
244:Q
234:Q
216:.
213:}
210:2
202:2
198:x
190:Q
183:x
180:{
177:=
174:S
161:Q
84:(
23:.
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