第 17509 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-epsilon-witness；資料={"limit":[8,9],"coefficients":[9,-10],"shift":2,"epsilon":[1,18],"point":0}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。

• G001 — 同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={"limit":[8,9],"coefficients":[9,-10],"shift":2,"epsilon":[1,18],"point":0}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 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=[8, 9];signed coefficients=[9, -10];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)

selected pointwise witness absolute coefficient cross product := mul(9, 18)

selected pointwise witness absolute coefficient cross product = 162

selected pointwise witness strict reciprocal threshold := div(162, 1)

selected pointwise witness strict reciprocal threshold = 162

selected pointwise witness raw strict integer cutoff := sum(162, -2, 1)

selected pointwise witness raw strict integer cutoff = 161

selected pointwise witness:point=0;SIGNED coefficient=9;absolute coefficient=9;positive epsilon=1/18;shift=2

threshold=162;floor=162;least nonnegative strict cutoff N=161

selected pointwise witness cutoff denominator := sum(161, 2)

selected pointwise witness cutoff denominator = 163

selected pointwise witness absolute error at cutoff := div(9, 163)

selected pointwise witness absolute error at cutoff = 9/163

selected pointwise witness previous denominator := sum(160, 2)

selected pointwise witness previous denominator = 162

selected pointwise witness previous absolute error := div(9, 162)

selected pointwise witness previous absolute error = 1/18

selected pointwise witness:current N=161;current error=9/163;previous error=1/18;epsilon=1/18

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

selected pointwise witness:declared exact tail;positive denominators increase for all n>=N=161

for EVERY later n,absolute error<=bound=9/163<epsilon=1/18

逐步反向核對

• 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 selected pointwise witness previous absolute error，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 selected pointwise witness previous denominator，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 selected pointwise witness 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 selected pointwise witness 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 selected pointwise witness 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 selected pointwise witness strict reciprocal threshold，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 selected pointwise witness 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

(0, 1/18, 161, 9/163, 是, forall_n_ge_N)

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。