第 17561 題：函數列逐點收斂

原題與已知條件

同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={"limit":[10,11],"coefficients":[11,-12],"shift":2,"epsilon":[1,20]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。

• G001 — 同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={"limit":[10,11],"coefficients":[11,-12],"shift":2,"epsilon":[1,20]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,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-pointwise-convergence-of-functions

逐步推導

same COMPLETE finite domain Fin(2);original aligned limits=[10, 11];signed coefficients=[11, -12];positive shift=2

declared for EVERY n>=0:f_n(x)=limit[x]+coefficient[x]/(n+shift);signed error=f_n(x)-limit[x]=coefficient[x]/(n+shift)

finite-domain witness point0 absolute coefficient cross product := mul(11, 20)

finite-domain witness point0 absolute coefficient cross product = 220

finite-domain witness point0 strict reciprocal threshold := div(220, 1)

finite-domain witness point0 strict reciprocal threshold = 220

finite-domain witness point0 raw strict integer cutoff := sum(220, -2, 1)

finite-domain witness point0 raw strict integer cutoff = 219

finite-domain witness point0:point=0;SIGNED coefficient=11;absolute coefficient=11;positive epsilon=1/20;shift=2

threshold=220;floor=220;least nonnegative strict cutoff N=219

finite-domain witness point0 cutoff denominator := sum(219, 2)

finite-domain witness point0 cutoff denominator = 221

finite-domain witness point0 absolute error at cutoff := div(11, 221)

finite-domain witness point0 absolute error at cutoff = 11/221

finite-domain witness point0 previous denominator := sum(218, 2)

finite-domain witness point0 previous denominator = 220

finite-domain witness point0 previous absolute error := div(11, 220)

finite-domain witness point0 previous absolute error = 1/20

finite-domain witness point0:current N=219;current error=11/221;previous error=1/20;epsilon=1/20

current strict bound holds;previous fails;all smaller indices fail by monotonicity

finite-domain witness point0:declared exact tail;positive denominators increase for all n>=N=219

for EVERY later n,absolute error<=bound=11/221<epsilon=1/20

finite-domain witness point1 absolute coefficient cross product := mul(12, 20)

finite-domain witness point1 absolute coefficient cross product = 240

finite-domain witness point1 strict reciprocal threshold := div(240, 1)

finite-domain witness point1 strict reciprocal threshold = 240

finite-domain witness point1 raw strict integer cutoff := sum(240, -2, 1)

finite-domain witness point1 raw strict integer cutoff = 239

finite-domain witness point1:point=1;SIGNED coefficient=-12;absolute coefficient=12;positive epsilon=1/20;shift=2

threshold=240;floor=240;least nonnegative strict cutoff N=239

finite-domain witness point1 cutoff denominator := sum(239, 2)

finite-domain witness point1 cutoff denominator = 241

finite-domain witness point1 absolute error at cutoff := div(12, 241)

finite-domain witness point1 absolute error at cutoff = 12/241

finite-domain witness point1 previous denominator := sum(238, 2)

finite-domain witness point1 previous denominator = 240

finite-domain witness point1 previous absolute error := div(12, 240)

finite-domain witness point1 previous absolute error = 1/20

finite-domain witness point1:current N=239;current error=12/241;previous error=1/20;epsilon=1/20

current strict bound holds;previous fails;all smaller indices fail by monotonicity

finite-domain witness point1:declared exact tail;positive denominators increase for all n>=N=239

for EVERY later n,absolute error<=bound=12/241<epsilon=1/20

finite-domain maximum witness denominator := sum(239, 2)

finite-domain maximum witness denominator = 241

finite-domain maximum absolute error point0 := div(11, 241)

finite-domain maximum absolute error point0 = 11/241

finite-domain maximum absolute error point1 := div(12, 241)

finite-domain maximum absolute error point1 = 12/241

positive epsilon=1/20;ALL pointwise witness rows=[[0, 219], [1, 239]];finite domain size=2

finite maximum N=239;all point errors at common N=[11/241, 12/241];same N works for every point and every later n

逐步反向核對

• 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 finite-domain maximum absolute error point1，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 finite-domain maximum absolute error point0，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 finite-domain maximum witness denominator，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 — 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 — 以精確有理數重算 finite-domain witness point1 previous absolute error，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 finite-domain witness point1 previous denominator，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 finite-domain witness point1 absolute error at cutoff，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 finite-domain witness point1 cutoff denominator，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 finite-domain witness point1 raw strict integer cutoff，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 finite-domain witness point1 strict reciprocal threshold，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 finite-domain witness point1 absolute coefficient cross product，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 finite-domain witness point0 previous absolute error，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 finite-domain witness point0 previous denominator，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 finite-domain witness point0 absolute error at cutoff，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 finite-domain witness point0 cutoff denominator，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 finite-domain witness point0 raw strict integer cutoff，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 finite-domain witness point0 strict reciprocal threshold，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 finite-domain witness point0 absolute coefficient cross product，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

(1/20, [[0, 219], [1, 239]], 239, 是, finite_domain_maximum_only_not_infinite_domain_theorem)

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。