第 21028 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":"increasing","limit":[3,4],"coefficient":3,"shift":2,"epsilon":[1,12]}。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":"increasing","limit":[3,4],"coefficient":3,"shift":2,"epsilon":[1,12]}。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=increasing,L=3/4,c=3>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)*(3)/((n+2)*(n+1+2));positive denominator;all a_n strictly below L=3/4

WHOLE increasing law with ORIGINAL L=3/4,c=3,shift=2

for ANY epsilon>0 choose natural N with N+2>3/epsilon;ALL n>=N have |a_n-L|<epsilon;L is supremum but NOT attained because c/(n+shift)>0

monoseq c over epsilon := div(3, 1/12)

monoseq c over epsilon = 36

monoseq index multiple := mul(1, 35)

monoseq index multiple = 35

monoseq actual source index := sum(0, 35)

monoseq actual source index = 35

monoseq witness denominator := sum(35, 2)

monoseq witness denominator = 37

monoseq witness distance := div(3, 37)

monoseq witness distance = 3/37

monoseq predecessor distance := div(3, 36)

monoseq predecessor distance = 1/12

monoseq:c=3,shift=2,epsilon=1/12;index(q)=1q+0;c/epsilon=36;floor=36

least nonnegative witness=35;actual source index=35;distance=3/37<epsilon;predecessor distance=1/12;no predecessor evaluated when witness0

monoseq:positive c=3,epsilon=1/12,shift=2,stride=1,offset=0;least witness=35

for EVERY integer q>=35,denominator=1q+0+2>=witness denominator>0,so 0<distance<=(3/37)<epsilon

monoseq at witness N denominator := sum(35, 2)

monoseq at witness N denominator = 37

monoseq at witness N unsigned tail := div(3, 37)

monoseq at witness N unsigned tail = 3/37

monoseq at witness N signed value := sub(3/4, 3/37)

monoseq at witness N signed value = 99/148

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

(increasing, 3/4, 1/12, 35, 99/148, 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。