1036:"}},"text\/plain":{"en":{"":"checksum"}}},"{\"value\":\"37be80fa29a24561c2b8b4ae53989c8c459c0ad3\",\"type\":\"string\"}":{"text\/html":{"en":{"P4092":"37be80fa29a24561c2b8b4ae53989c8c459c0ad3"}},"text\/plain":{"en":{"P4092":"37be80fa29a24561c2b8b4ae53989c8c459c0ad3"}}},"{\"value\":{\"entity-type\":\"property\",\"numeric-id\":459,\"id\":\"P459\"},\"type\":\"wikibase-entityid\"}":{"text\/html":{"en":{"":"
613:
601:
589:
577:
565:
553:
541:
43:
960:"}},"text\/plain":{"en":{"":"inception"}}},"{\"value\":{\"time\":\"+2016-02-04T00:00:00Z\",\"timezone\":0,\"before\":0,\"after\":0,\"precision\":11,\"calendarmodel\":\"http:\\\/\\\/www.wikidata.org\\\/entity\\\/Q1985727\"},\"type\":\"time\"}":{"text\/html":{"en":{"P571":"4 February 2016"}},"text\/plain":{"en":{"P571":"4 February 2016"}}}}": -->
697:
1032:
708:
657:
95:
636:
998:"}},"text\/plain":{"en":{"":"media type"}}},"{\"value\":\"image\\\/svg+xml\",\"type\":\"string\"}":{"text\/html":{"en":{"P1163":"image\/svg+xml"}},"text\/plain":{"en":{"P1163":"image\/svg+xml"}}}}": -->
1366:
1336:
1306:
1040:"}},"text\/plain":{"en":{"":"determination method"}}},"{\"value\":{\"entity-type\":\"item\",\"numeric-id\":13414952,\"id\":\"Q13414952\"},\"type\":\"wikibase-entityid\"}":{"text\/html":{"en":{"P459":"
872:"}},"text\/plain":{"en":{"":"copyright license"}}},"{\"value\":{\"entity-type\":\"item\",\"numeric-id\":18199165,\"id\":\"Q18199165\"},\"type\":\"wikibase-entityid\"}":{"text\/html":{"en":{"P275":"
828:"}},"text\/plain":{"en":{"":"copyright status"}}},"{\"value\":{\"entity-type\":\"item\",\"numeric-id\":50423863,\"id\":\"Q50423863\"},\"type\":\"wikibase-entityid\"}":{"text\/html":{"en":{"P6216":"
916:"}},"text\/plain":{"en":{"":"source of file"}}},"{\"value\":{\"entity-type\":\"item\",\"numeric-id\":66458942,\"id\":\"Q66458942\"},\"type\":\"wikibase-entityid\"}":{"text\/html":{"en":{"P7482":"
1103:"}},"text\/plain":{"en":{"":"data size"}}},"{\"value\":{\"amount\":\"+25422\",\"unit\":\"http:\\\/\\\/www.wikidata.org\\\/entity\\\/Q8799\"},\"type\":\"quantity\"}":{"text\/html":{"en":{"P3575":"25,422
1148:"}},"text\/plain":{"en":{"":"height"}}},"{\"value\":{\"amount\":\"+107\",\"unit\":\"http:\\\/\\\/www.wikidata.org\\\/entity\\\/Q355198\"},\"type\":\"quantity\"}":{"text\/html":{"en":{"P2048":"107
1193:"}},"text\/plain":{"en":{"":"width"}}},"{\"value\":{\"amount\":\"+552\",\"unit\":\"http:\\\/\\\/www.wikidata.org\\\/entity\\\/Q355198\"},\"type\":\"quantity\"}":{"text\/html":{"en":{"P2049":"552
1276:
1189:
647:
1450:
749:– You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
69:
65:
61:
57:
53:
47:
1041:
917:
873:
829:
1440:
1190:
1145:
1100:
1033:
995:
913:
825:
488:
453:
361:
180:
1037:
957:
869:
268:
106:
420:
390:
328:
298:
78:
220:
200:
876:"}},"text\/plain":{"en":{"P275":"Creative Commons Attribution-ShareAlike 4.0 International"}}}}": -->
612:
600:
588:
576:
564:
552:
540:
1365:
1360:
1335:
1330:
1305:
1300:
1085:
943:
899:
855:
1473:
This file contains additional information, probably added from the digital camera or scanner used to create or digitize it.
756:
457:
With continuing this process we get nested intervals. The point being in all those intervals is an accumulation point of
1204:
1159:
1114:
1051:
1005:
927:
839:
920:"}},"text\/plain":{"en":{"P7482":"original creation by uploader"}}}}": -->
1078:
967:
883:
814:
1275:
1270:
42:
1397:
718:
1476:
If the file has been modified from its original state, some details may not fully reflect the modified file.
1425:
755:– If you remix, transform, or build upon the material, you must distribute your contributions under the
1107:"}},"text\/plain":{"en":{"P3575":"25,422 byte"}}}}": -->
832:"}},"text\/plain":{"en":{"P6216":"copyrighted"}}}}": -->
1393:
The following pages on the
English Knowledge use this file (pages on other projects are not listed):
1197:"}},"text\/plain":{"en":{"P2049":"552 pixel"}}}}": -->
1152:"}},"text\/plain":{"en":{"P2048":"107 pixel"}}}}": -->
114:
1044:"}},"text\/plain":{"en":{"P459":"SHA-1"}}}}": -->
460:
425:
333:
152:
715:
228:
398:
368:
306:
276:
8:
525:
1430:
1376:
1346:
1316:
1286:
205:
185:
101:
804:
690:
I, the copyright holder of this work, hereby publish it under the following license:
663:
1415:
1214:
1169:
1124:
1061:
1015:
977:
937:
893:
849:
667:
642:
1243:
Click on a date/time to view the file as it appeared at that time.
656:
635:
900:
Creative
Commons Attribution-ShareAlike 4.0 International
874:
Creative
Commons Attribution-ShareAlike 4.0 International
501:
89:
83:(SVG file, nominally 552 × 107 pixels, file size: 25 KB)
798:
Add a one-line explanation of what this file represents
1391:
463:
428:
401:
371:
336:
309:
279:
231:
208:
188:
155:
1406:
1241:
182:
on the real number line with one of its lower bound
113:
Commons is a freely licensed media file repository.
1426:
Mathe für Nicht-Freaks: Satz von
Bolzano-Weierstraß
521:
482:
447:
414:
384:
355:
322:
292:
262:
214:
194:
174:
766:https://creativecommons.org/licenses/by-sa/4.0
94:
772:Creative Commons Attribution-Share Alike 4.0
696:
497:
136:
532:
517:
507:
330:, which contains infinitely many members of
14:
1194:
1149:
1104:
732:– to copy, distribute and transmit the work
46:Size of this PNG preview of this SVG file:
1431:Serlo: EN: The Bolzano-Weierstrass theorem
422:which contains infinitely many members of
392:again into two equally sized subintervals.
1408:The following other wikis use this file:
719:Attribution-Share Alike 4.0 International
1067:37be80fa29a24561c2b8b4ae53989c8c459c0ad3
139:Bolzano–Weierstrass theorem - step 1.svg
618:7. step of Bolzano–Weierstrass theorem
606:6. step of Bolzano–Weierstrass theorem
594:5. step of Bolzano–Weierstrass theorem
582:4. step of Bolzano–Weierstrass theorem
570:3. step of Bolzano–Weierstrass theorem
558:2. step of Bolzano–Weierstrass theorem
546:1. step of Bolzano–Weierstrass theorem
125:
300:into twi equally sized subintervals.
1383:User created page with UploadWizard
1234:
707:
702:
13:
1471:
803:
797:
693:
689:
661:
640:
143:
126:
76:
1505:
1478:
802:
1392:
788:
743:Under the following conditions:
714:This file is licensed under the
706:
701:
695:
655:
634:
611:
599:
587:
575:
563:
551:
539:
93:
31:
21:
785:
144:
1461:ทฤษฎีบทบ็อลท์ซาโน-ไวเออร์ชตราส
1403:
1238:
784:
477:
464:
442:
429:
350:
337:
257:
245:
169:
156:
26:
1:
944:original creation by uploader
918:original creation by uploader
303:We take this subinterval for
36:
1388:
810:Items portrayed in this file
683:
7:
1468:
1416:Satz von Bolzano-Weierstraß
1398:Bolzano–Weierstrass theorem
225:We take the first interval
202:and one of its upper bound
10:
1510:
1457:Usage on th.wikipedia.org
1447:Usage on ja.wikipedia.org
1437:Usage on fa.wikipedia.org
1422:Usage on de.wikibooks.org
1412:Usage on de.wikipedia.org
757:same or compatible license
273:Now we split the interval
133:
128:
1235:
511:
395:We take this subinterval
16:
787:
723:
671:, or with something else
100:This is a file from the
41:
1382:
1352:
1322:
1292:
500:
483:{\displaystyle (x_{n})}
448:{\displaystyle (x_{n})}
356:{\displaystyle (x_{n})}
175:{\displaystyle (x_{n})}
104:. Information from its
1441:قضیه بولزانو-وایرشتراس
1361:16:24, 4 February 2016
1331:17:53, 4 February 2016
1301:17:55, 4 February 2016
1271:20:05, 4 February 2016
484:
449:
416:
386:
357:
324:
294:
264:
263:{\displaystyle I_{1}=}
216:
196:
176:
107:description page there
485:
450:
417:
415:{\displaystyle I_{3}}
387:
385:{\displaystyle I_{2}}
358:
325:
323:{\displaystyle I_{2}}
295:
293:{\displaystyle I_{1}}
265:
217:
197:
177:
149:# A bounded sequence
1079:determination method
1038:determination method
626:SVG development
461:
426:
399:
369:
334:
307:
277:
229:
206:
186:
153:
1451:ボルツァーノ＝ワイエルシュトラスの定理
738:– to adapt the work
52:Other resolutions:
526:User:Stephan Kulla
480:
445:
412:
382:
353:
320:
290:
260:
212:
192:
172:
70:2,560 × 496 pixels
66:1,280 × 248 pixels
62:1,024 × 198 pixels
1497:
1496:
1404:Global file usage
1386:
1196:
1151:
1106:
884:copyright license
870:copyright license
796:
680:
679:
666:was created with
215:{\displaystyle S}
195:{\displaystyle s}
122:
121:
102:Wikimedia Commons
32:Global file usage
1501:
1479:
1373:
1343:
1313:
1283:
1245:
1223:
1178:
1133:
1088:
1083:
840:copyright status
826:copyright status
794:
779:
776:
773:
770:
767:
759:as the original.
716:Creative Commons
710:
709:
705:
704:
699:
698:
659:
650:
638:
615:
603:
591:
579:
567:
555:
543:
513:
503:
489:
487:
486:
481:
476:
475:
454:
452:
451:
446:
441:
440:
421:
419:
418:
413:
411:
410:
391:
389:
388:
383:
381:
380:
362:
360:
359:
354:
349:
348:
329:
327:
326:
321:
319:
318:
299:
297:
296:
291:
289:
288:
269:
267:
266:
261:
241:
240:
221:
219:
218:
213:
201:
199:
198:
193:
181:
179:
178:
173:
168:
167:
148:
140:
134:
118:
97:
96:
90:
84:
73:
58:640 × 124 pixels
48:552 × 107 pixels
1509:
1508:
1504:
1503:
1502:
1500:
1499:
1498:
1467:
1402:
1394:
1387:
1379:
1371:
1349:
1341:
1319:
1311:
1289:
1281:
1237:
1236:
1233:
1232:
1231:
1230:
1229:
1228:
1227:
1226:
1224:
1221:
1211:
1210:
1209:
1207:
1188:
1187:
1186:
1185:
1184:
1183:
1182:
1181:
1179:
1176:
1166:
1165:
1164:
1162:
1143:
1142:
1141:
1140:
1139:
1138:
1137:
1136:
1134:
1131:
1121:
1120:
1119:
1117:
1098:
1097:
1096:
1095:
1094:
1093:
1092:
1091:
1090:
1089:
1084:
1081:
1072:
1071:
1070:
1068:
1058:
1057:
1056:
1054:
1031:
1030:
1029:
1028:
1027:
1026:
1025:
1024:
1022:
1012:
1011:
1010:
1008:
993:
992:
991:
990:
989:
988:
987:
986:
984:
983:4 February 2016
974:
973:
972:
970:
955:
954:
953:
952:
951:
950:
949:
948:
946:
934:
933:
932:
930:
911:
910:
909:
908:
907:
906:
905:
904:
902:
890:
889:
888:
886:
867:
866:
865:
864:
863:
862:
861:
860:
858:
846:
845:
844:
842:
823:
822:
821:
820:
819:
817:
801:
800:
799:
782:
781:
780:
777:
774:
771:
768:
765:
764:
722:
711:
692:
691:
686:
681:
676:
675:
674:
653:
652:
646:
629:
619:
616:
607:
604:
595:
592:
583:
580:
571:
568:
559:
556:
547:
544:
502:4 February 2016
493:
471:
467:
462:
459:
458:
436:
432:
427:
424:
423:
406:
402:
400:
397:
396:
376:
372:
370:
367:
366:
344:
340:
335:
332:
331:
314:
310:
308:
305:
304:
284:
280:
278:
275:
274:
236:
232:
230:
227:
226:
207:
204:
203:
187:
184:
183:
163:
159:
154:
151:
150:
138:
131:
124:
123:
112:
111:
110:is shown below.
86:
82:
75:
74:
54:320 × 62 pixels
51:
12:
11:
5:
1507:
1495:
1494:
1491:
1487:
1486:
1483:
1470:
1466:
1465:
1464:
1463:
1455:
1454:
1453:
1445:
1444:
1443:
1435:
1434:
1433:
1428:
1420:
1419:
1418:
1405:
1401:
1400:
1390:
1385:
1384:
1381:
1377:
1374:
1368:
1363:
1358:
1355:
1354:
1351:
1347:
1344:
1338:
1333:
1328:
1325:
1324:
1321:
1317:
1314:
1308:
1303:
1298:
1295:
1294:
1291:
1287:
1284:
1278:
1273:
1268:
1264:
1263:
1260:
1257:
1254:
1251:
1248:
1240:
1225:
1219:
1217:
1216:
1215:
1213:
1212:
1208:
1203:
1201:
1200:
1199:
1198:
1192:
1180:
1174:
1172:
1171:
1170:
1168:
1167:
1163:
1158:
1156:
1155:
1154:
1153:
1147:
1135:
1129:
1127:
1126:
1125:
1123:
1122:
1118:
1113:
1111:
1110:
1109:
1108:
1102:
1077:
1076:
1075:
1074:
1073:
1069:
1066:
1064:
1063:
1062:
1060:
1059:
1055:
1050:
1048:
1047:
1046:
1045:
1043:
1039:
1035:
1023:
1020:
1018:
1017:
1016:
1014:
1013:
1009:
1004:
1002:
1001:
1000:
999:
997:
985:
982:
980:
979:
978:
976:
975:
971:
966:
964:
963:
962:
961:
959:
947:
942:
940:
939:
938:
936:
935:
931:
928:source of file
926:
924:
923:
922:
921:
919:
915:
914:source of file
903:
898:
896:
895:
894:
892:
891:
887:
882:
880:
879:
878:
877:
875:
871:
859:
854:
852:
851:
850:
848:
847:
843:
838:
836:
835:
834:
833:
831:
827:
818:
813:
811:
808:
807:
806:
805:
793:
792:
789:
786:
763:
762:
761:
760:
750:
741:
740:
739:
733:
726:You are free:
713:
712:
694:
688:
687:
685:
682:
678:
677:
654:
633:
632:
630:
627:
623:
622:
621:
620:
617:
610:
608:
605:
598:
596:
593:
586:
584:
581:
574:
572:
569:
562:
560:
557:
550:
548:
545:
538:
534:
533:Other versions
530:
529:
519:
515:
514:
509:
505:
504:
499:
495:
494:
492:
491:
479:
474:
470:
466:
455:
444:
439:
435:
431:
409:
405:
393:
379:
375:
365:Then we split
363:
352:
347:
343:
339:
317:
313:
301:
287:
283:
271:
259:
256:
253:
250:
247:
244:
239:
235:
211:
191:
171:
166:
162:
158:
141:
132:
130:
127:
120:
119:
98:
88:
87:
45:
40:
39:
34:
29:
24:
19:
9:
6:
4:
3:
2:
1506:
1492:
1489:
1488:
1484:
1481:
1480:
1477:
1474:
1462:
1459:
1458:
1456:
1452:
1449:
1448:
1446:
1442:
1439:
1438:
1436:
1432:
1429:
1427:
1424:
1423:
1421:
1417:
1414:
1413:
1411:
1410:
1409:
1399:
1396:
1395:
1380:
1378:Stephan Kulla
1375:
1369:
1367:
1364:
1362:
1359:
1357:
1356:
1350:
1348:Stephan Kulla
1345:
1339:
1337:
1334:
1332:
1329:
1327:
1326:
1323:change border
1320:
1318:Stephan Kulla
1315:
1309:
1307:
1304:
1302:
1299:
1297:
1296:
1290:
1288:Stephan Kulla
1285:
1279:
1277:
1274:
1272:
1269:
1266:
1265:
1261:
1258:
1255:
1252:
1249:
1247:
1246:
1244:
1218:
1206:
1202:
1173:
1161:
1157:
1128:
1116:
1112:
1087:
1080:
1065:
1053:
1049:
1021:image/svg+xml
1019:
1007:
1003:
981:
969:
965:
945:
941:
929:
925:
901:
897:
885:
881:
857:
853:
841:
837:
816:
812:
809:
790:
783:
769:CC BY-SA 4.0
758:
754:
751:
748:
745:
744:
742:
737:
734:
731:
728:
727:
725:
724:
720:
717:
700:
672:
669:
665:
658:
649:
644:
637:
631:
625:
624:
614:
609:
602:
597:
590:
585:
578:
573:
566:
561:
554:
549:
542:
537:
536:
535:
531:
527:
523:
522:Stephan Kulla
520:
516:
510:
506:
496:
472:
468:
456:
437:
433:
407:
403:
394:
377:
373:
364:
345:
341:
315:
311:
302:
285:
281:
272:
254:
251:
248:
242:
237:
233:
224:
223:
209:
189:
164:
160:
147:
142:
135:
116:
109:
108:
103:
99:
92:
91:
85:
80:
79:Original file
71:
67:
63:
59:
55:
49:
44:
38:
35:
33:
30:
28:
25:
23:
20:
18:
15:
1475:
1472:
1407:
1242:
1239:File history
752:
746:
735:
729:
670:
664:vector image
145:
115:You can help
105:
77:
22:File history
1493:30.080751mm
1485:155.90843mm
1353:new version
1293:new version
856:copyrighted
830:copyrighted
753:share alike
747:attribution
137:Description
1389:File usage
1370:552 × 107
1340:552 × 149
1310:552 × 107
1280:552 × 107
1256:Dimensions
1006:media type
996:media type
27:File usage
1253:Thumbnail
1250:Date/Time
1115:data size
1101:data size
968:inception
958:inception
684:Licensing
628:InfoField
146:English:
1469:Metadata
1052:checksum
1034:checksum
791:Captions
736:to remix
730:to share
721:license.
668:Inkscape
645:code is
512:Own work
37:Metadata
1372:(19 KB)
1342:(21 KB)
1312:(21 KB)
1282:(25 KB)
1267:current
1262:Comment
1130:25,422
815:depicts
795:English
129:Summary
81:
1490:Height
1160:height
1146:height
662:This
660:
639:
518:Author
508:Source
1482:Width
1222:pixel
1205:width
1195:pixel
1191:width
1177:pixel
1150:pixel
1086:SHA-1
1042:SHA-1
648:valid
1259:User
1220:552
1175:107
1132:byte
1105:byte
778:true
775:true
641:The
498:Date
17:File
643:SVG
1082::
222:.
68:|
64:|
60:|
56:|
50:.
673:.
651:.
528:)
524:(
490:.
478:)
473:n
469:x
465:(
443:)
438:n
434:x
430:(
408:3
404:I
378:2
374:I
351:)
346:n
342:x
338:(
316:2
312:I
286:1
282:I
270:.
258:]
255:S
252:,
249:s
246:[
243:=
238:1
234:I
210:S
190:s
170:)
165:n
161:x
157:(
117:.
72:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
