第 12133 題：Cauchy序列

原題與已知條件

保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={"sequence":[[1,9],[1,10],[1,11],[1,12],[1,13]],"epsilon":[1,5],"N":1}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。

• G001 — 保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={"sequence":[[1,9],[1,10],[1,11],[1,12],[1,13]],"epsilon":[1,5],"N":1}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-cauchy-sequence

逐步推導

raw sequence=[[1, 9], [1, 10], [1, 11], [1, 12], [1, 13]];raw epsilon=[1, 5]

normalized sequence=[1/9, 1/10, 1/11, 1/12, 1/13];positive epsilon=1/5;cutoff=1

recorded pair1,1 signed difference := sub(1/10, 1/10)

recorded pair1,1 signed difference = 0

ordered recorded indices=[1, 1];signed difference=0;epsilon=1/5

absolute distance=0;strictly below epsilon=是

recorded pair1,2 signed difference := sub(1/10, 1/11)

recorded pair1,2 signed difference = 1/110

ordered recorded indices=[1, 2];signed difference=1/110;epsilon=1/5

absolute distance=1/110;strictly below epsilon=是

recorded pair1,3 signed difference := sub(1/10, 1/12)

recorded pair1,3 signed difference = 1/60

ordered recorded indices=[1, 3];signed difference=1/60;epsilon=1/5

absolute distance=1/60;strictly below epsilon=是

recorded pair1,4 signed difference := sub(1/10, 1/13)

recorded pair1,4 signed difference = 3/130

ordered recorded indices=[1, 4];signed difference=3/130;epsilon=1/5

absolute distance=3/130;strictly below epsilon=是

recorded pair2,1 signed difference := sub(1/11, 1/10)

recorded pair2,1 signed difference = -1/110

ordered recorded indices=[2, 1];signed difference=-1/110;epsilon=1/5

absolute distance=1/110;strictly below epsilon=是

recorded pair2,2 signed difference := sub(1/11, 1/11)

recorded pair2,2 signed difference = 0

ordered recorded indices=[2, 2];signed difference=0;epsilon=1/5

absolute distance=0;strictly below epsilon=是

recorded pair2,3 signed difference := sub(1/11, 1/12)

recorded pair2,3 signed difference = 1/132

ordered recorded indices=[2, 3];signed difference=1/132;epsilon=1/5

absolute distance=1/132;strictly below epsilon=是

recorded pair2,4 signed difference := sub(1/11, 1/13)

recorded pair2,4 signed difference = 2/143

ordered recorded indices=[2, 4];signed difference=2/143;epsilon=1/5

absolute distance=2/143;strictly below epsilon=是

recorded pair3,1 signed difference := sub(1/12, 1/10)

recorded pair3,1 signed difference = -1/60

ordered recorded indices=[3, 1];signed difference=-1/60;epsilon=1/5

absolute distance=1/60;strictly below epsilon=是

recorded pair3,2 signed difference := sub(1/12, 1/11)

recorded pair3,2 signed difference = -1/132

ordered recorded indices=[3, 2];signed difference=-1/132;epsilon=1/5

absolute distance=1/132;strictly below epsilon=是

recorded pair3,3 signed difference := sub(1/12, 1/12)

recorded pair3,3 signed difference = 0

ordered recorded indices=[3, 3];signed difference=0;epsilon=1/5

absolute distance=0;strictly below epsilon=是

recorded pair3,4 signed difference := sub(1/12, 1/13)

recorded pair3,4 signed difference = 1/156

ordered recorded indices=[3, 4];signed difference=1/156;epsilon=1/5

absolute distance=1/156;strictly below epsilon=是

recorded pair4,1 signed difference := sub(1/13, 1/10)

recorded pair4,1 signed difference = -3/130

ordered recorded indices=[4, 1];signed difference=-3/130;epsilon=1/5

absolute distance=3/130;strictly below epsilon=是

recorded pair4,2 signed difference := sub(1/13, 1/11)

recorded pair4,2 signed difference = -2/143

ordered recorded indices=[4, 2];signed difference=-2/143;epsilon=1/5

absolute distance=2/143;strictly below epsilon=是

recorded pair4,3 signed difference := sub(1/13, 1/12)

recorded pair4,3 signed difference = -1/156

ordered recorded indices=[4, 3];signed difference=-1/156;epsilon=1/5

absolute distance=1/156;strictly below epsilon=是

recorded pair4,4 signed difference := sub(1/13, 1/13)

recorded pair4,4 signed difference = 0

ordered recorded indices=[4, 4];signed difference=0;epsilon=1/5

absolute distance=0;strictly below epsilon=是

all bad recorded pairs=[]

finite recorded tail passes=是;general Cauchy established=否

逐步反向核對

• S034 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S033 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S032 · premise_and_result_audit — 以精確有理數重算 recorded pair4,4 signed difference，不以小數近似替代。
• S031 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S030 · premise_and_result_audit — 以精確有理數重算 recorded pair4,3 signed difference，不以小數近似替代。
• S029 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S028 · premise_and_result_audit — 以精確有理數重算 recorded pair4,2 signed difference，不以小數近似替代。
• S027 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S026 · premise_and_result_audit — 以精確有理數重算 recorded pair4,1 signed difference，不以小數近似替代。
• S025 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S024 · premise_and_result_audit — 以精確有理數重算 recorded pair3,4 signed difference，不以小數近似替代。
• S023 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S022 · premise_and_result_audit — 以精確有理數重算 recorded pair3,3 signed difference，不以小數近似替代。
• S021 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S020 · premise_and_result_audit — 以精確有理數重算 recorded pair3,2 signed difference，不以小數近似替代。
• S019 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S018 · premise_and_result_audit — 以精確有理數重算 recorded pair3,1 signed difference，不以小數近似替代。
• S017 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S016 · premise_and_result_audit — 以精確有理數重算 recorded pair2,4 signed difference，不以小數近似替代。
• S015 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S014 · premise_and_result_audit — 以精確有理數重算 recorded pair2,3 signed difference，不以小數近似替代。
• S013 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S012 · premise_and_result_audit — 以精確有理數重算 recorded pair2,2 signed difference，不以小數近似替代。
• S011 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S010 · premise_and_result_audit — 以精確有理數重算 recorded pair2,1 signed difference，不以小數近似替代。
• S009 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S008 · premise_and_result_audit — 以精確有理數重算 recorded pair1,4 signed difference，不以小數近似替代。
• S007 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S006 · premise_and_result_audit — 以精確有理數重算 recorded pair1,3 signed difference，不以小數近似替代。
• S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S004 · premise_and_result_audit — 以精確有理數重算 recorded pair1,2 signed difference，不以小數近似替代。
• S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S002 · premise_and_result_audit — 以精確有理數重算 recorded pair1,1 signed difference，不以小數近似替代。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

([1/9, 1/10, 1/11, 1/12, 1/13], 1/5, 1, [], 是, 否, 有限尾段不足以證一般Cauchy)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。