第 12132 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":[[1,8],[1,9],[1,10],[1,11],[1,12]],"epsilon":[1,4],"N":0}。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":[[1,8],[1,9],[1,10],[1,11],[1,12]],"epsilon":[1,4],"N":0}。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, 8], [1, 9], [1, 10], [1, 11], [1, 12]];raw epsilon=[1, 4]

normalized sequence=[1/8, 1/9, 1/10, 1/11, 1/12];positive epsilon=1/4;cutoff=0

recorded pair0,0 signed difference := sub(1/8, 1/8)

recorded pair0,0 signed difference = 0

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

absolute distance=0;strictly below epsilon=是

recorded pair0,1 signed difference := sub(1/8, 1/9)

recorded pair0,1 signed difference = 1/72

ordered recorded indices=[0, 1];signed difference=1/72;epsilon=1/4

absolute distance=1/72;strictly below epsilon=是

recorded pair0,2 signed difference := sub(1/8, 1/10)

recorded pair0,2 signed difference = 1/40

ordered recorded indices=[0, 2];signed difference=1/40;epsilon=1/4

absolute distance=1/40;strictly below epsilon=是

recorded pair0,3 signed difference := sub(1/8, 1/11)

recorded pair0,3 signed difference = 3/88

ordered recorded indices=[0, 3];signed difference=3/88;epsilon=1/4

absolute distance=3/88;strictly below epsilon=是

recorded pair0,4 signed difference := sub(1/8, 1/12)

recorded pair0,4 signed difference = 1/24

ordered recorded indices=[0, 4];signed difference=1/24;epsilon=1/4

absolute distance=1/24;strictly below epsilon=是

recorded pair1,0 signed difference := sub(1/9, 1/8)

recorded pair1,0 signed difference = -1/72

ordered recorded indices=[1, 0];signed difference=-1/72;epsilon=1/4

absolute distance=1/72;strictly below epsilon=是

recorded pair1,1 signed difference := sub(1/9, 1/9)

recorded pair1,1 signed difference = 0

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

absolute distance=0;strictly below epsilon=是

recorded pair1,2 signed difference := sub(1/9, 1/10)

recorded pair1,2 signed difference = 1/90

ordered recorded indices=[1, 2];signed difference=1/90;epsilon=1/4

absolute distance=1/90;strictly below epsilon=是

recorded pair1,3 signed difference := sub(1/9, 1/11)

recorded pair1,3 signed difference = 2/99

ordered recorded indices=[1, 3];signed difference=2/99;epsilon=1/4

absolute distance=2/99;strictly below epsilon=是

recorded pair1,4 signed difference := sub(1/9, 1/12)

recorded pair1,4 signed difference = 1/36

ordered recorded indices=[1, 4];signed difference=1/36;epsilon=1/4

absolute distance=1/36;strictly below epsilon=是

recorded pair2,0 signed difference := sub(1/10, 1/8)

recorded pair2,0 signed difference = -1/40

ordered recorded indices=[2, 0];signed difference=-1/40;epsilon=1/4

absolute distance=1/40;strictly below epsilon=是

recorded pair2,1 signed difference := sub(1/10, 1/9)

recorded pair2,1 signed difference = -1/90

ordered recorded indices=[2, 1];signed difference=-1/90;epsilon=1/4

absolute distance=1/90;strictly below epsilon=是

recorded pair2,2 signed difference := sub(1/10, 1/10)

recorded pair2,2 signed difference = 0

ordered recorded indices=[2, 2];signed difference=0;epsilon=1/4

absolute distance=0;strictly below epsilon=是

recorded pair2,3 signed difference := sub(1/10, 1/11)

recorded pair2,3 signed difference = 1/110

ordered recorded indices=[2, 3];signed difference=1/110;epsilon=1/4

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

recorded pair2,4 signed difference := sub(1/10, 1/12)

recorded pair2,4 signed difference = 1/60

ordered recorded indices=[2, 4];signed difference=1/60;epsilon=1/4

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

recorded pair3,0 signed difference := sub(1/11, 1/8)

recorded pair3,0 signed difference = -3/88

ordered recorded indices=[3, 0];signed difference=-3/88;epsilon=1/4

absolute distance=3/88;strictly below epsilon=是

recorded pair3,1 signed difference := sub(1/11, 1/9)

recorded pair3,1 signed difference = -2/99

ordered recorded indices=[3, 1];signed difference=-2/99;epsilon=1/4

absolute distance=2/99;strictly below epsilon=是

recorded pair3,2 signed difference := sub(1/11, 1/10)

recorded pair3,2 signed difference = -1/110

ordered recorded indices=[3, 2];signed difference=-1/110;epsilon=1/4

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

recorded pair3,3 signed difference := sub(1/11, 1/11)

recorded pair3,3 signed difference = 0

ordered recorded indices=[3, 3];signed difference=0;epsilon=1/4

absolute distance=0;strictly below epsilon=是

recorded pair3,4 signed difference := sub(1/11, 1/12)

recorded pair3,4 signed difference = 1/132

ordered recorded indices=[3, 4];signed difference=1/132;epsilon=1/4

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

recorded pair4,0 signed difference := sub(1/12, 1/8)

recorded pair4,0 signed difference = -1/24

ordered recorded indices=[4, 0];signed difference=-1/24;epsilon=1/4

absolute distance=1/24;strictly below epsilon=是

recorded pair4,1 signed difference := sub(1/12, 1/9)

recorded pair4,1 signed difference = -1/36

ordered recorded indices=[4, 1];signed difference=-1/36;epsilon=1/4

absolute distance=1/36;strictly below epsilon=是

recorded pair4,2 signed difference := sub(1/12, 1/10)

recorded pair4,2 signed difference = -1/60

ordered recorded indices=[4, 2];signed difference=-1/60;epsilon=1/4

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

recorded pair4,3 signed difference := sub(1/12, 1/11)

recorded pair4,3 signed difference = -1/132

ordered recorded indices=[4, 3];signed difference=-1/132;epsilon=1/4

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

recorded pair4,4 signed difference := sub(1/12, 1/12)

recorded pair4,4 signed difference = 0

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

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

• S052 · 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。
• S051 · 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。
• S050 · 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。
• S049 · 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。
• S048 · 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。
• S047 · 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。
• S046 · 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。
• S045 · 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。
• S044 · 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。
• S043 · 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。
• S042 · premise_and_result_audit — 以精確有理數重算 recorded pair4,0 signed difference，不以小數近似替代。
• S041 · 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。
• S040 · 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。
• S039 · 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。
• S038 · 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。
• S037 · 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。
• S036 · 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。
• S035 · 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。
• S034 · 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。
• 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 pair3,0 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 pair2,4 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 pair2,3 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 pair2,2 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 pair2,1 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 pair2,0 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 pair1,4 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 pair1,3 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 pair1,2 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 pair1,1 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 pair1,0 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 pair0,4 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 pair0,3 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 pair0,2 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 pair0,1 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 pair0,0 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

([1/8, 1/9, 1/10, 1/11, 1/12], 1/4, 0, [], 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。