第 21033 題：單調有界數列收斂定理

原題與已知條件

實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={"direction":"decreasing","limit":[8,9],"coefficient":8,"shift":2,"epsilon":[1,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={"direction":"decreasing","limit":[8,9],"coefficient":8,"shift":2,"epsilon":[1,17]}。請依 exact_ordered_tuple_declared_by_interface 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-monotone-sequence-convergence-theorem

逐步推導

source operation=monoseq-epsilon-witness;original snapshot and mathematics retained

explicit ordered source output fields=[direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate]

ORIGINAL direction=decreasing,L=8/9,c=8>0,shift=2>=1;WHOLE a_n=L+(1)c/(n+shift) for ALL n>=0

for EVERY n>=0,a_(n+1)-a_n=(-1)*(8)/((n+2)*(n+1+2));positive denominator;all a_n strictly above L=8/9

WHOLE decreasing law with ORIGINAL L=8/9,c=8,shift=2

for ANY epsilon>0 choose natural N with N+2>8/epsilon;ALL n>=N have |a_n-L|<epsilon;L is infimum but NOT attained because c/(n+shift)>0

monoseq c over epsilon := div(8, 1/17)

monoseq c over epsilon = 136

monoseq index multiple := mul(1, 135)

monoseq index multiple = 135

monoseq actual source index := sum(0, 135)

monoseq actual source index = 135

monoseq witness denominator := sum(135, 2)

monoseq witness denominator = 137

monoseq witness distance := div(8, 137)

monoseq witness distance = 8/137

monoseq predecessor distance := div(8, 136)

monoseq predecessor distance = 1/17

monoseq:c=8,shift=2,epsilon=1/17;index(q)=1q+0;c/epsilon=136;floor=136

least nonnegative witness=135;actual source index=135;distance=8/137<epsilon;predecessor distance=1/17;no predecessor evaluated when witness0

monoseq:positive c=8,epsilon=1/17,shift=2,stride=1,offset=0;least witness=135

for EVERY integer q>=135,denominator=1q+0+2>=witness denominator>0,so 0<distance<=(8/137)<epsilon

monoseq at witness N denominator := sum(135, 2)

monoseq at witness N denominator = 137

monoseq at witness N unsigned tail := div(8, 137)

monoseq at witness N unsigned tail = 8/137

monoseq at witness N signed value := sum(8/9, 8/137)

monoseq at witness N signed value = 1168/1233

逐步反向核對

• S014 · premise_and_result_audit — 以精確有理數重算 monoseq at witness N signed value，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 monoseq at witness N unsigned tail，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 monoseq at witness N denominator，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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S009 · premise_and_result_audit — 以精確有理數重算 monoseq predecessor distance，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 monoseq witness distance，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 monoseq witness denominator，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 monoseq actual source index，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 monoseq index multiple，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 monoseq c over epsilon，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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• 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

(decreasing, 8/9, 1/17, 135, 1168/1233, 是, forall_n_ge_N_by_monotone_symbolic_tail)

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。