# 實數列Bolzano－Weierstrass定理基礎 · 脈衝 R02

本批已通過 Schema、去重、來源快照與獨立答案重算；仍未標記為人工審定。

本題庫已有對應的中英逐步詳解。原題的 candidate 與英文 ai_candidate 狀態保留；AI 覆核、有限計算與 Lean 形式證明分開標示。

batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations · 2026-10-14 · 100 題

欄序為來源綁定的呈現補充；原始 JSONL 與數學答案不變。

[欄序與來源綁定資料](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={"period_values":[1,-2,[3,2]]}。請依 [period_values, absolute_bound, whole_sequence_bounded, certificate] 欄序回答。

查看答案與解釋([1, -2, 3/2], 2, 是, whole_infinite_periodic_sequence_bounded)

Wave6第三層核對=([1, -2, 3/2], 2, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0001/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-boundedness · candidate





## 2. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={"period_values":[2,-3,[4,2]]}。請依 [period_values, absolute_bound, whole_sequence_bounded, certificate] 欄序回答。

查看答案與解釋([2, -3, 2], 3, 是, whole_infinite_periodic_sequence_bounded)

Wave6第三層核對=([2, -3, 2], 3, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0002/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-boundedness · candidate





## 3. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={"period_values":[3,-4,[5,2]]}。請依 [period_values, absolute_bound, whole_sequence_bounded, certificate] 欄序回答。

查看答案與解釋([3, -4, 5/2], 4, 是, whole_infinite_periodic_sequence_bounded)

Wave6第三層核對=([3, -4, 5/2], 4, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0003/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-boundedness · candidate





## 4. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={"period_values":[4,-5,[6,2]]}。請依 [period_values, absolute_bound, whole_sequence_bounded, certificate] 欄序回答。

查看答案與解釋([4, -5, 3], 5, 是, whole_infinite_periodic_sequence_bounded)

Wave6第三層核對=([4, -5, 3], 5, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0004/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-boundedness · candidate





## 5. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={"period_values":[5,-6,[7,2]]}。請依 [period_values, absolute_bound, whole_sequence_bounded, certificate] 欄序回答。

查看答案與解釋([5, -6, 7/2], 6, 是, whole_infinite_periodic_sequence_bounded)

Wave6第三層核對=([5, -6, 7/2], 6, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0005/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-boundedness · candidate





## 6. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={"period_values":[6,-7,[8,2]]}。請依 [period_values, absolute_bound, whole_sequence_bounded, certificate] 欄序回答。

查看答案與解釋([6, -7, 4], 7, 是, whole_infinite_periodic_sequence_bounded)

Wave6第三層核對=([6, -7, 4], 7, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0006/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-boundedness · candidate





## 7. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={"period_values":[7,-8,[9,2]]}。請依 [period_values, absolute_bound, whole_sequence_bounded, certificate] 欄序回答。

查看答案與解釋([7, -8, 9/2], 8, 是, whole_infinite_periodic_sequence_bounded)

Wave6第三層核對=([7, -8, 9/2], 8, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0007/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-boundedness · candidate





## 8. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={"period_values":[8,-9,[10,2]]}。請依 [period_values, absolute_bound, whole_sequence_bounded, certificate] 欄序回答。

查看答案與解釋([8, -9, 5], 9, 是, whole_infinite_periodic_sequence_bounded)

Wave6第三層核對=([8, -9, 5], 9, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0008/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-boundedness · candidate





## 9. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={"period_values":[9,-10,[11,2]]}。請依 [period_values, absolute_bound, whole_sequence_bounded, certificate] 欄序回答。

查看答案與解釋([9, -10, 11/2], 10, 是, whole_infinite_periodic_sequence_bounded)

Wave6第三層核對=([9, -10, 11/2], 10, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0009/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-boundedness · candidate





## 10. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={"period_values":[10,-11,[12,2]]}。請依 [period_values, absolute_bound, whole_sequence_bounded, certificate] 欄序回答。

查看答案與解釋([10, -11, 6], 11, 是, whole_infinite_periodic_sequence_bounded)

Wave6第三層核對=([10, -11, 6], 11, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0010/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-boundedness · candidate





## 11. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={"period_values":[11,-12,[13,2]]}。請依 [period_values, absolute_bound, whole_sequence_bounded, certificate] 欄序回答。

查看答案與解釋([11, -12, 13/2], 12, 是, whole_infinite_periodic_sequence_bounded)

Wave6第三層核對=([11, -12, 13/2], 12, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0011/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-boundedness · candidate





## 12. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={"period_values":[12,-13,[14,2]]}。請依 [period_values, absolute_bound, whole_sequence_bounded, certificate] 欄序回答。

查看答案與解釋([12, -13, 7], 13, 是, whole_infinite_periodic_sequence_bounded)

Wave6第三層核對=([12, -13, 7], 13, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0012/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-boundedness · candidate





## 13. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={"period_values":[13,-14,[15,2]]}。請依 [period_values, absolute_bound, whole_sequence_bounded, certificate] 欄序回答。

查看答案與解釋([13, -14, 15/2], 14, 是, whole_infinite_periodic_sequence_bounded)

Wave6第三層核對=([13, -14, 15/2], 14, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0013/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-boundedness · candidate





## 14. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={"period_values":[14,-15,[16,2]]}。請依 [period_values, absolute_bound, whole_sequence_bounded, certificate] 欄序回答。

查看答案與解釋([14, -15, 8], 15, 是, whole_infinite_periodic_sequence_bounded)

Wave6第三層核對=([14, -15, 8], 15, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0014/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-boundedness · candidate





## 15. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={"period_values":[15,-16,[17,2]]}。請依 [period_values, absolute_bound, whole_sequence_bounded, certificate] 欄序回答。

查看答案與解釋([15, -16, 17/2], 16, 是, whole_infinite_periodic_sequence_bounded)

Wave6第三層核對=([15, -16, 17/2], 16, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0015/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-boundedness · candidate





## 16. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={"period_values":[16,-17,[18,2]]}。請依 [period_values, absolute_bound, whole_sequence_bounded, certificate] 欄序回答。

查看答案與解釋([16, -17, 9], 17, 是, whole_infinite_periodic_sequence_bounded)

Wave6第三層核對=([16, -17, 9], 17, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0016/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-boundedness · candidate





## 17. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={"period_values":[17,-18,[19,2]]}。請依 [period_values, absolute_bound, whole_sequence_bounded, certificate] 欄序回答。

查看答案與解釋([17, -18, 19/2], 18, 是, whole_infinite_periodic_sequence_bounded)

Wave6第三層核對=([17, -18, 19/2], 18, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0017/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-boundedness · candidate





## 18. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={"period_values":[18,-19,[20,2]]}。請依 [period_values, absolute_bound, whole_sequence_bounded, certificate] 欄序回答。

查看答案與解釋([18, -19, 10], 19, 是, whole_infinite_periodic_sequence_bounded)

Wave6第三層核對=([18, -19, 10], 19, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0018/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-boundedness · candidate





## 19. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={"period_values":[19,-20,[21,2]]}。請依 [period_values, absolute_bound, whole_sequence_bounded, certificate] 欄序回答。

查看答案與解釋([19, -20, 21/2], 20, 是, whole_infinite_periodic_sequence_bounded)

Wave6第三層核對=([19, -20, 21/2], 20, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0019/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-boundedness · candidate





## 20. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={"period_values":[20,-21,[22,2]]}。請依 [period_values, absolute_bound, whole_sequence_bounded, certificate] 欄序回答。

查看答案與解釋([20, -21, 11], 21, 是, whole_infinite_periodic_sequence_bounded)

Wave6第三層核對=([20, -21, 11], 21, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0020/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-boundedness · candidate





## 21. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={"period_values":[21,-22,[23,2]]}。請依 [period_values, absolute_bound, whole_sequence_bounded, certificate] 欄序回答。

查看答案與解釋([21, -22, 23/2], 22, 是, whole_infinite_periodic_sequence_bounded)

Wave6第三層核對=([21, -22, 23/2], 22, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0021/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-boundedness · candidate





## 22. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={"period_values":[22,-23,[24,2]]}。請依 [period_values, absolute_bound, whole_sequence_bounded, certificate] 欄序回答。

查看答案與解釋([22, -23, 12], 23, 是, whole_infinite_periodic_sequence_bounded)

Wave6第三層核對=([22, -23, 12], 23, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0022/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-boundedness · candidate





## 23. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={"period_values":[23,-24,[25,2]]}。請依 [period_values, absolute_bound, whole_sequence_bounded, certificate] 欄序回答。

查看答案與解釋([23, -24, 25/2], 24, 是, whole_infinite_periodic_sequence_bounded)

Wave6第三層核對=([23, -24, 25/2], 24, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0023/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-boundedness · candidate





## 24. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={"period_values":[24,-25,[26,2]]}。請依 [period_values, absolute_bound, whole_sequence_bounded, certificate] 欄序回答。

查看答案與解釋([24, -25, 13], 25, 是, whole_infinite_periodic_sequence_bounded)

Wave6第三層核對=([24, -25, 13], 25, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0024/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-boundedness · candidate





## 25. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={"period_values":[25,-26,[27,2]]}。請依 [period_values, absolute_bound, whole_sequence_bounded, certificate] 欄序回答。

查看答案與解釋([25, -26, 27/2], 26, 是, whole_infinite_periodic_sequence_bounded)

Wave6第三層核對=([25, -26, 27/2], 26, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0025/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-boundedness · candidate





## 26. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={"period_values":[1,-2,[3,2]],"residue":0,"j":1}。請依 [residue, j, k_j, value, constant_subsequence_certificate, certificate] 欄序回答。

查看答案與解釋(0, 1, 3, 1, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

Wave6第三層核對=(0, 1, 3, 1, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0026/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-subsequence · candidate





## 27. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={"period_values":[2,-3,[4,2]],"residue":1,"j":2}。請依 [residue, j, k_j, value, constant_subsequence_certificate, certificate] 欄序回答。

查看答案與解釋(1, 2, 7, -3, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

Wave6第三層核對=(1, 2, 7, -3, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0027/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-subsequence · candidate





## 28. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={"period_values":[3,-4,[5,2]],"residue":2,"j":3}。請依 [residue, j, k_j, value, constant_subsequence_certificate, certificate] 欄序回答。

查看答案與解釋(2, 3, 11, 5/2, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

Wave6第三層核對=(2, 3, 11, 5/2, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0028/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-subsequence · candidate





## 29. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={"period_values":[4,-5,[6,2]],"residue":0,"j":4}。請依 [residue, j, k_j, value, constant_subsequence_certificate, certificate] 欄序回答。

查看答案與解釋(0, 4, 12, 4, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

Wave6第三層核對=(0, 4, 12, 4, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0029/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-subsequence · candidate





## 30. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={"period_values":[5,-6,[7,2]],"residue":1,"j":5}。請依 [residue, j, k_j, value, constant_subsequence_certificate, certificate] 欄序回答。

查看答案與解釋(1, 5, 16, -6, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

Wave6第三層核對=(1, 5, 16, -6, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0030/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-subsequence · candidate





## 31. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={"period_values":[6,-7,[8,2]],"residue":2,"j":6}。請依 [residue, j, k_j, value, constant_subsequence_certificate, certificate] 欄序回答。

查看答案與解釋(2, 6, 20, 4, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

Wave6第三層核對=(2, 6, 20, 4, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0031/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-subsequence · candidate





## 32. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={"period_values":[7,-8,[9,2]],"residue":0,"j":7}。請依 [residue, j, k_j, value, constant_subsequence_certificate, certificate] 欄序回答。

查看答案與解釋(0, 7, 21, 7, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

Wave6第三層核對=(0, 7, 21, 7, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0032/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-subsequence · candidate





## 33. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={"period_values":[8,-9,[10,2]],"residue":1,"j":8}。請依 [residue, j, k_j, value, constant_subsequence_certificate, certificate] 欄序回答。

查看答案與解釋(1, 8, 25, -9, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

Wave6第三層核對=(1, 8, 25, -9, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0033/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-subsequence · candidate





## 34. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={"period_values":[9,-10,[11,2]],"residue":2,"j":9}。請依 [residue, j, k_j, value, constant_subsequence_certificate, certificate] 欄序回答。

查看答案與解釋(2, 9, 29, 11/2, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

Wave6第三層核對=(2, 9, 29, 11/2, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0034/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-subsequence · candidate





## 35. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={"period_values":[10,-11,[12,2]],"residue":0,"j":10}。請依 [residue, j, k_j, value, constant_subsequence_certificate, certificate] 欄序回答。

查看答案與解釋(0, 10, 30, 10, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

Wave6第三層核對=(0, 10, 30, 10, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0035/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-subsequence · candidate





## 36. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={"period_values":[11,-12,[13,2]],"residue":1,"j":11}。請依 [residue, j, k_j, value, constant_subsequence_certificate, certificate] 欄序回答。

查看答案與解釋(1, 11, 34, -12, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

Wave6第三層核對=(1, 11, 34, -12, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0036/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-subsequence · candidate





## 37. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={"period_values":[12,-13,[14,2]],"residue":2,"j":12}。請依 [residue, j, k_j, value, constant_subsequence_certificate, certificate] 欄序回答。

查看答案與解釋(2, 12, 38, 7, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

Wave6第三層核對=(2, 12, 38, 7, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0037/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-subsequence · candidate





## 38. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={"period_values":[13,-14,[15,2]],"residue":0,"j":13}。請依 [residue, j, k_j, value, constant_subsequence_certificate, certificate] 欄序回答。

查看答案與解釋(0, 13, 39, 13, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

Wave6第三層核對=(0, 13, 39, 13, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0038/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-subsequence · candidate





## 39. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={"period_values":[14,-15,[16,2]],"residue":1,"j":14}。請依 [residue, j, k_j, value, constant_subsequence_certificate, certificate] 欄序回答。

查看答案與解釋(1, 14, 43, -15, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

Wave6第三層核對=(1, 14, 43, -15, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0039/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-subsequence · candidate





## 40. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={"period_values":[15,-16,[17,2]],"residue":2,"j":15}。請依 [residue, j, k_j, value, constant_subsequence_certificate, certificate] 欄序回答。

查看答案與解釋(2, 15, 47, 17/2, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

Wave6第三層核對=(2, 15, 47, 17/2, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0040/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-subsequence · candidate





## 41. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={"period_values":[16,-17,[18,2]],"residue":0,"j":16}。請依 [residue, j, k_j, value, constant_subsequence_certificate, certificate] 欄序回答。

查看答案與解釋(0, 16, 48, 16, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

Wave6第三層核對=(0, 16, 48, 16, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0041/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-subsequence · candidate





## 42. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={"period_values":[17,-18,[19,2]],"residue":1,"j":17}。請依 [residue, j, k_j, value, constant_subsequence_certificate, certificate] 欄序回答。

查看答案與解釋(1, 17, 52, -18, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

Wave6第三層核對=(1, 17, 52, -18, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0042/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-subsequence · candidate





## 43. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={"period_values":[18,-19,[20,2]],"residue":2,"j":18}。請依 [residue, j, k_j, value, constant_subsequence_certificate, certificate] 欄序回答。

查看答案與解釋(2, 18, 56, 10, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

Wave6第三層核對=(2, 18, 56, 10, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0043/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-subsequence · candidate





## 44. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={"period_values":[19,-20,[21,2]],"residue":0,"j":19}。請依 [residue, j, k_j, value, constant_subsequence_certificate, certificate] 欄序回答。

查看答案與解釋(0, 19, 57, 19, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

Wave6第三層核對=(0, 19, 57, 19, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0044/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-subsequence · candidate





## 45. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={"period_values":[20,-21,[22,2]],"residue":1,"j":20}。請依 [residue, j, k_j, value, constant_subsequence_certificate, certificate] 欄序回答。

查看答案與解釋(1, 20, 61, -21, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

Wave6第三層核對=(1, 20, 61, -21, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0045/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-subsequence · candidate





## 46. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={"period_values":[21,-22,[23,2]],"residue":2,"j":21}。請依 [residue, j, k_j, value, constant_subsequence_certificate, certificate] 欄序回答。

查看答案與解釋(2, 21, 65, 23/2, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

Wave6第三層核對=(2, 21, 65, 23/2, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0046/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-subsequence · candidate





## 47. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={"period_values":[22,-23,[24,2]],"residue":0,"j":22}。請依 [residue, j, k_j, value, constant_subsequence_certificate, certificate] 欄序回答。

查看答案與解釋(0, 22, 66, 22, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

Wave6第三層核對=(0, 22, 66, 22, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0047/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-subsequence · candidate





## 48. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={"period_values":[23,-24,[25,2]],"residue":1,"j":23}。請依 [residue, j, k_j, value, constant_subsequence_certificate, certificate] 欄序回答。

查看答案與解釋(1, 23, 70, -24, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

Wave6第三層核對=(1, 23, 70, -24, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0048/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-subsequence · candidate





## 49. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={"period_values":[24,-25,[26,2]],"residue":2,"j":24}。請依 [residue, j, k_j, value, constant_subsequence_certificate, certificate] 欄序回答。

查看答案與解釋(2, 24, 74, 13, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

Wave6第三層核對=(2, 24, 74, 13, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0049/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-subsequence · candidate





## 50. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={"period_values":[25,-26,[27,2]],"residue":0,"j":25}。請依 [residue, j, k_j, value, constant_subsequence_certificate, certificate] 欄序回答。

查看答案與解釋(0, 25, 75, 25, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

Wave6第三層核對=(0, 25, 75, 25, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0050/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-periodic-subsequence · candidate





## 51. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[1,2],"coefficient":1,"shift":2,"parity":0,"epsilon":[1,10]}。請依 [parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate] 欄序回答。

查看答案與解釋(0, k(j)=2j+0, 1/2, 1/10, 5, 7/12, 是, forall_j_ge_J_symbolic_subsequence_tail)

Wave6第三層核對=(0, k(j)=2j+0, 1/2, 1/10, 5, 7/12, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0051/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-oscillatory-tail · candidate





## 52. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[2,3],"coefficient":2,"shift":3,"parity":1,"epsilon":[1,11]}。請依 [parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate] 欄序回答。

查看答案與解釋(1, k(j)=2j+1, 2/3, 1/11, 10, 7/12, 是, forall_j_ge_J_symbolic_subsequence_tail)

Wave6第三層核對=(1, k(j)=2j+1, 2/3, 1/11, 10, 7/12, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0052/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-oscillatory-tail · candidate





## 53. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[3,4],"coefficient":3,"shift":4,"parity":0,"epsilon":[1,12]}。請依 [parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate] 欄序回答。

查看答案與解釋(0, k(j)=2j+0, 3/4, 1/12, 17, 63/76, 是, forall_j_ge_J_symbolic_subsequence_tail)

Wave6第三層核對=(0, k(j)=2j+0, 3/4, 1/12, 17, 63/76, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0053/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-oscillatory-tail · candidate





## 54. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[4,5],"coefficient":4,"shift":5,"parity":1,"epsilon":[1,13]}。請依 [parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate] 欄序回答。

查看答案與解釋(1, k(j)=2j+1, 4/5, 1/13, 24, 98/135, 是, forall_j_ge_J_symbolic_subsequence_tail)

Wave6第三層核對=(1, k(j)=2j+1, 4/5, 1/13, 24, 98/135, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0054/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-oscillatory-tail · candidate





## 55. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[5,6],"coefficient":5,"shift":6,"parity":0,"epsilon":[1,14]}。請依 [parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate] 欄序回答。

查看答案與解釋(0, k(j)=2j+0, 5/6, 1/14, 33, 65/72, 是, forall_j_ge_J_symbolic_subsequence_tail)

Wave6第三層核對=(0, k(j)=2j+0, 5/6, 1/14, 33, 65/72, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0055/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-oscillatory-tail · candidate





## 56. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[6,7],"coefficient":6,"shift":7,"parity":1,"epsilon":[1,15]}。請依 [parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate] 欄序回答。

查看答案與解釋(1, k(j)=2j+1, 6/7, 1/15, 42, 255/322, 是, forall_j_ge_J_symbolic_subsequence_tail)

Wave6第三層核對=(1, k(j)=2j+1, 6/7, 1/15, 42, 255/322, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0056/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-oscillatory-tail · candidate





## 57. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[7,8],"coefficient":7,"shift":8,"parity":0,"epsilon":[1,16]}。請依 [parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate] 欄序回答。

查看答案與解釋(0, k(j)=2j+0, 7/8, 1/16, 53, 427/456, 是, forall_j_ge_J_symbolic_subsequence_tail)

Wave6第三層核對=(0, k(j)=2j+0, 7/8, 1/16, 53, 427/456, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0057/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-oscillatory-tail · candidate





## 58. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[8,9],"coefficient":8,"shift":9,"parity":1,"epsilon":[1,17]}。請依 [parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate] 欄序回答。

查看答案與解釋(1, k(j)=2j+1, 8/9, 1/17, 64, 172/207, 是, forall_j_ge_J_symbolic_subsequence_tail)

Wave6第三層核對=(1, k(j)=2j+1, 8/9, 1/17, 64, 172/207, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0058/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-oscillatory-tail · candidate





## 59. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[9,10],"coefficient":9,"shift":10,"parity":0,"epsilon":[1,18]}。請依 [parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate] 欄序回答。

查看答案與解釋(0, k(j)=2j+0, 9/10, 1/18, 77, 783/820, 是, forall_j_ge_J_symbolic_subsequence_tail)

Wave6第三層核對=(0, k(j)=2j+0, 9/10, 1/18, 77, 783/820, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0059/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-oscillatory-tail · candidate





## 60. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[10,11],"coefficient":10,"shift":11,"parity":1,"epsilon":[1,19]}。請依 [parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate] 欄序回答。

查看答案與解釋(1, k(j)=2j+1, 10/11, 1/19, 90, 905/1056, 是, forall_j_ge_J_symbolic_subsequence_tail)

Wave6第三層核對=(1, k(j)=2j+1, 10/11, 1/19, 90, 905/1056, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0060/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-oscillatory-tail · candidate





## 61. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[11,12],"coefficient":11,"shift":12,"parity":0,"epsilon":[1,20]}。請依 [parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate] 欄序回答。

查看答案與解釋(0, k(j)=2j+0, 11/12, 1/20, 105, 143/148, 是, forall_j_ge_J_symbolic_subsequence_tail)

Wave6第三層核對=(0, k(j)=2j+0, 11/12, 1/20, 105, 143/148, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0061/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-oscillatory-tail · candidate





## 62. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[12,13],"coefficient":12,"shift":13,"parity":1,"epsilon":[1,21]}。請依 [parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate] 欄序回答。

查看答案與解釋(1, k(j)=2j+1, 12/13, 1/21, 120, 1446/1651, 是, forall_j_ge_J_symbolic_subsequence_tail)

Wave6第三層核對=(1, k(j)=2j+1, 12/13, 1/21, 120, 1446/1651, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0062/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-oscillatory-tail · candidate





## 63. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[13,14],"coefficient":13,"shift":14,"parity":0,"epsilon":[1,22]}。請依 [parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate] 欄序回答。

查看答案與解釋(0, k(j)=2j+0, 13/14, 1/22, 137, 1963/2016, 是, forall_j_ge_J_symbolic_subsequence_tail)

Wave6第三層核對=(0, k(j)=2j+0, 13/14, 1/22, 137, 1963/2016, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0063/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-oscillatory-tail · candidate





## 64. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[14,15],"coefficient":14,"shift":15,"parity":1,"epsilon":[1,23]}。請依 [parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate] 欄序回答。

查看答案與解釋(1, k(j)=2j+1, 14/15, 1/23, 154, 721/810, 是, forall_j_ge_J_symbolic_subsequence_tail)

Wave6第三層核對=(1, k(j)=2j+1, 14/15, 1/23, 154, 721/810, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0064/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-oscillatory-tail · candidate





## 65. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[15,16],"coefficient":15,"shift":16,"parity":0,"epsilon":[1,24]}。請依 [parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate] 欄序回答。

查看答案與解釋(0, k(j)=2j+0, 15/16, 1/24, 173, 2835/2896, 是, forall_j_ge_J_symbolic_subsequence_tail)

Wave6第三層核對=(0, k(j)=2j+0, 15/16, 1/24, 173, 2835/2896, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0065/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-oscillatory-tail · candidate





## 66. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[16,17],"coefficient":16,"shift":17,"parity":1,"epsilon":[1,25]}。請依 [parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate] 欄序回答。

查看答案與解釋(1, k(j)=2j+1, 16/17, 1/25, 192, 3080/3417, 是, forall_j_ge_J_symbolic_subsequence_tail)

Wave6第三層核對=(1, k(j)=2j+1, 16/17, 1/25, 192, 3080/3417, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0066/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-oscillatory-tail · candidate





## 67. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[17,18],"coefficient":17,"shift":18,"parity":0,"epsilon":[1,26]}。請依 [parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate] 欄序回答。

查看答案與解釋(0, k(j)=2j+0, 17/18, 1/26, 213, 1309/1332, 是, forall_j_ge_J_symbolic_subsequence_tail)

Wave6第三層核對=(0, k(j)=2j+0, 17/18, 1/26, 213, 1309/1332, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0067/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-oscillatory-tail · candidate





## 68. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[18,19],"coefficient":18,"shift":19,"parity":1,"epsilon":[1,27]}。請依 [parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate] 欄序回答。

查看答案與解釋(1, k(j)=2j+1, 18/19, 1/27, 234, 4221/4636, 是, forall_j_ge_J_symbolic_subsequence_tail)

Wave6第三層核對=(1, k(j)=2j+1, 18/19, 1/27, 234, 4221/4636, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0068/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-oscillatory-tail · candidate





## 69. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[19,20],"coefficient":19,"shift":20,"parity":0,"epsilon":[1,28]}。請依 [parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate] 欄序回答。

查看答案與解釋(0, k(j)=2j+0, 19/20, 1/28, 257, 5263/5340, 是, forall_j_ge_J_symbolic_subsequence_tail)

Wave6第三層核對=(0, k(j)=2j+0, 19/20, 1/28, 257, 5263/5340, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0069/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-oscillatory-tail · candidate





## 70. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[20,21],"coefficient":20,"shift":21,"parity":1,"epsilon":[1,29]}。請依 [parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate] 欄序回答。

查看答案與解釋(1, k(j)=2j+1, 20/21, 1/29, 280, 1870/2037, 是, forall_j_ge_J_symbolic_subsequence_tail)

Wave6第三層核對=(1, k(j)=2j+1, 20/21, 1/29, 280, 1870/2037, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0070/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-oscillatory-tail · candidate





## 71. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[21,22],"coefficient":21,"shift":22,"parity":0,"epsilon":[1,30]}。請依 [parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate] 欄序回答。

查看答案與解釋(0, k(j)=2j+0, 21/22, 1/30, 305, 6867/6952, 是, forall_j_ge_J_symbolic_subsequence_tail)

Wave6第三層核對=(0, k(j)=2j+0, 21/22, 1/30, 305, 6867/6952, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0071/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-oscillatory-tail · candidate





## 72. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[22,23],"coefficient":22,"shift":23,"parity":1,"epsilon":[1,31]}。請依 [parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate] 欄序回答。

查看答案與解釋(1, k(j)=2j+1, 22/23, 1/31, 330, 7271/7866, 是, forall_j_ge_J_symbolic_subsequence_tail)

Wave6第三層核對=(1, k(j)=2j+1, 22/23, 1/31, 330, 7271/7866, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0072/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-oscillatory-tail · candidate





## 73. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[23,24],"coefficient":23,"shift":24,"parity":0,"epsilon":[1,32]}。請依 [parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate] 欄序回答。

查看答案與解釋(0, k(j)=2j+0, 23/24, 1/32, 357, 2921/2952, 是, forall_j_ge_J_symbolic_subsequence_tail)

Wave6第三層核對=(0, k(j)=2j+0, 23/24, 1/32, 357, 2921/2952, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0073/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-oscillatory-tail · candidate





## 74. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[24,25],"coefficient":24,"shift":25,"parity":1,"epsilon":[1,33]}。請依 [parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate] 欄序回答。

查看答案與解釋(1, k(j)=2j+1, 24/25, 1/33, 384, 9228/9925, 是, forall_j_ge_J_symbolic_subsequence_tail)

Wave6第三層核對=(1, k(j)=2j+1, 24/25, 1/33, 384, 9228/9925, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0074/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-oscillatory-tail · candidate





## 75. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[25,26],"coefficient":25,"shift":26,"parity":0,"epsilon":[1,34]}。請依 [parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate] 欄序回答。

查看答案與解釋(0, k(j)=2j+0, 25/26, 1/34, 413, 10975/11076, 是, forall_j_ge_J_symbolic_subsequence_tail)

Wave6第三層核對=(0, k(j)=2j+0, 25/26, 1/34, 413, 10975/11076, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0075/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-oscillatory-tail · candidate





## 76. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={"values":[0,1,0,2]}。請依 [prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary] 欄序回答。

三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。

查看答案與解釋([0, 1, 0, 2], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

Wave6第三層核對=([0, 1, 0, 2], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0076/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-finite-prefix-boundary · candidate





## 77. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={"values":[1,2,1,3]}。請依 [prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary] 欄序回答。

三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。

查看答案與解釋([1, 2, 1, 3], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

Wave6第三層核對=([1, 2, 1, 3], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0077/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-finite-prefix-boundary · candidate





## 78. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={"values":[2,3,2,4]}。請依 [prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary] 欄序回答。

三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。

查看答案與解釋([2, 3, 2, 4], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

Wave6第三層核對=([2, 3, 2, 4], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0078/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-finite-prefix-boundary · candidate





## 79. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={"values":[3,4,3,5]}。請依 [prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary] 欄序回答。

三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。

查看答案與解釋([3, 4, 3, 5], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

Wave6第三層核對=([3, 4, 3, 5], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0079/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-finite-prefix-boundary · candidate





## 80. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={"values":[4,5,4,6]}。請依 [prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary] 欄序回答。

三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。

查看答案與解釋([4, 5, 4, 6], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

Wave6第三層核對=([4, 5, 4, 6], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0080/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-finite-prefix-boundary · candidate





## 81. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={"values":[5,6,5,7]}。請依 [prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary] 欄序回答。

三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。

查看答案與解釋([5, 6, 5, 7], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

Wave6第三層核對=([5, 6, 5, 7], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0081/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-finite-prefix-boundary · candidate





## 82. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={"values":[6,7,6,8]}。請依 [prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary] 欄序回答。

三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。

查看答案與解釋([6, 7, 6, 8], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

Wave6第三層核對=([6, 7, 6, 8], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0082/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-finite-prefix-boundary · candidate





## 83. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={"values":[7,8,7,9]}。請依 [prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary] 欄序回答。

三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。

查看答案與解釋([7, 8, 7, 9], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

Wave6第三層核對=([7, 8, 7, 9], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0083/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-finite-prefix-boundary · candidate





## 84. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={"values":[8,9,8,10]}。請依 [prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary] 欄序回答。

三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。

查看答案與解釋([8, 9, 8, 10], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

Wave6第三層核對=([8, 9, 8, 10], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0084/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-finite-prefix-boundary · candidate





## 85. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={"values":[9,10,9,11]}。請依 [prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary] 欄序回答。

三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。

查看答案與解釋([9, 10, 9, 11], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

Wave6第三層核對=([9, 10, 9, 11], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0085/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-finite-prefix-boundary · candidate





## 86. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={"values":[10,11,10,12]}。請依 [prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary] 欄序回答。

三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。

查看答案與解釋([10, 11, 10, 12], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

Wave6第三層核對=([10, 11, 10, 12], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0086/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-finite-prefix-boundary · candidate





## 87. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={"values":[11,12,11,13]}。請依 [prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary] 欄序回答。

三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。

查看答案與解釋([11, 12, 11, 13], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

Wave6第三層核對=([11, 12, 11, 13], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0087/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-finite-prefix-boundary · candidate





## 88. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={"values":[12,13,12,14]}。請依 [prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary] 欄序回答。

三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。

查看答案與解釋([12, 13, 12, 14], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

Wave6第三層核對=([12, 13, 12, 14], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0088/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-finite-prefix-boundary · candidate





## 89. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={"values":[13,14,13,15]}。請依 [prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary] 欄序回答。

三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。

查看答案與解釋([13, 14, 13, 15], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

Wave6第三層核對=([13, 14, 13, 15], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0089/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-finite-prefix-boundary · candidate





## 90. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={"values":[14,15,14,16]}。請依 [prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary] 欄序回答。

三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。

查看答案與解釋([14, 15, 14, 16], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

Wave6第三層核對=([14, 15, 14, 16], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0090/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-finite-prefix-boundary · candidate





## 91. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={"values":[15,16,15,17]}。請依 [prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary] 欄序回答。

三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。

查看答案與解釋([15, 16, 15, 17], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

Wave6第三層核對=([15, 16, 15, 17], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0091/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-finite-prefix-boundary · candidate





## 92. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={"values":[16,17,16,18]}。請依 [prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary] 欄序回答。

三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。

查看答案與解釋([16, 17, 16, 18], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

Wave6第三層核對=([16, 17, 16, 18], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0092/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-finite-prefix-boundary · candidate





## 93. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={"values":[17,18,17,19]}。請依 [prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary] 欄序回答。

三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。

查看答案與解釋([17, 18, 17, 19], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

Wave6第三層核對=([17, 18, 17, 19], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0093/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-finite-prefix-boundary · candidate





## 94. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={"values":[18,19,18,20]}。請依 [prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary] 欄序回答。

三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。

查看答案與解釋([18, 19, 18, 20], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

Wave6第三層核對=([18, 19, 18, 20], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0094/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-finite-prefix-boundary · candidate





## 95. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={"values":[19,20,19,21]}。請依 [prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary] 欄序回答。

三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。

查看答案與解釋([19, 20, 19, 21], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

Wave6第三層核對=([19, 20, 19, 21], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0095/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-finite-prefix-boundary · candidate





## 96. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={"values":[20,21,20,22]}。請依 [prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary] 欄序回答。

三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。

查看答案與解釋([20, 21, 20, 22], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

Wave6第三層核對=([20, 21, 20, 22], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0096/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-finite-prefix-boundary · candidate





## 97. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={"values":[21,22,21,23]}。請依 [prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary] 欄序回答。

三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。

查看答案與解釋([21, 22, 21, 23], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

Wave6第三層核對=([21, 22, 21, 23], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0097/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-finite-prefix-boundary · candidate





## 98. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={"values":[22,23,22,24]}。請依 [prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary] 欄序回答。

三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。

查看答案與解釋([22, 23, 22, 24], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

Wave6第三層核對=([22, 23, 22, 24], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0098/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-finite-prefix-boundary · candidate





## 99. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={"values":[23,24,23,25]}。請依 [prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary] 欄序回答。

三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。

查看答案與解釋([23, 24, 23, 25], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

Wave6第三層核對=([23, 24, 23, 25], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0099/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-finite-prefix-boundary · candidate





## 100. 實數列Bolzano－Weierstrass定理

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={"values":[24,25,24,26]}。請依 [prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary] 欄序回答。

三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。

查看答案與解釋([24, 25, 24, 26], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

Wave6第三層核對=([24, 25, 24, 26], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0100/)

實數列Bolzano－Weierstrass定理 · qf-mod40-bw-finite-prefix-boundary · candidate



[原始題目 JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations/questions.jsonl) · [驗收報告](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations/validation-report.json)