第 12140 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":[[1,16],[1,17],[1,18],[1,19],[1,20]],"epsilon":[1,7],"N":2}。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":[[1,16],[1,17],[1,18],[1,19],[1,20]],"epsilon":[1,7],"N":2}。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, 16], [1, 17], [1, 18], [1, 19], [1, 20]];raw epsilon=[1, 7]

normalized sequence=[1/16, 1/17, 1/18, 1/19, 1/20];positive epsilon=1/7;cutoff=2

recorded pair2,2 signed difference := sub(1/18, 1/18)

recorded pair2,2 signed difference = 0

ordered recorded indices=[2, 2];signed difference=0;epsilon=1/7

absolute distance=0;strictly below epsilon=是

recorded pair2,3 signed difference := sub(1/18, 1/19)

recorded pair2,3 signed difference = 1/342

ordered recorded indices=[2, 3];signed difference=1/342;epsilon=1/7

absolute distance=1/342;strictly below epsilon=是

recorded pair2,4 signed difference := sub(1/18, 1/20)

recorded pair2,4 signed difference = 1/180

ordered recorded indices=[2, 4];signed difference=1/180;epsilon=1/7

absolute distance=1/180;strictly below epsilon=是

recorded pair3,2 signed difference := sub(1/19, 1/18)

recorded pair3,2 signed difference = -1/342

ordered recorded indices=[3, 2];signed difference=-1/342;epsilon=1/7

absolute distance=1/342;strictly below epsilon=是

recorded pair3,3 signed difference := sub(1/19, 1/19)

recorded pair3,3 signed difference = 0

ordered recorded indices=[3, 3];signed difference=0;epsilon=1/7

absolute distance=0;strictly below epsilon=是

recorded pair3,4 signed difference := sub(1/19, 1/20)

recorded pair3,4 signed difference = 1/380

ordered recorded indices=[3, 4];signed difference=1/380;epsilon=1/7

absolute distance=1/380;strictly below epsilon=是

recorded pair4,2 signed difference := sub(1/20, 1/18)

recorded pair4,2 signed difference = -1/180

ordered recorded indices=[4, 2];signed difference=-1/180;epsilon=1/7

absolute distance=1/180;strictly below epsilon=是

recorded pair4,3 signed difference := sub(1/20, 1/19)

recorded pair4,3 signed difference = -1/380

ordered recorded indices=[4, 3];signed difference=-1/380;epsilon=1/7

absolute distance=1/380;strictly below epsilon=是

recorded pair4,4 signed difference := sub(1/20, 1/20)

recorded pair4,4 signed difference = 0

ordered recorded indices=[4, 4];signed difference=0;epsilon=1/7

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

• S020 · 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。
• 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 pair4,4 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 pair4,3 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 pair4,2 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 pair3,4 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 pair3,3 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 pair3,2 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 pair2,4 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 pair2,3 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 pair2,2 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

([1/16, 1/17, 1/18, 1/19, 1/20], 1/7, 2, [], 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。