# 第 17524 題：函數列逐點收斂 [返回所在分頁 88](https://ocme.evemisslab.com/zh-Hant/questions/page/88/) [← 逐題詳解目錄](https://ocme.evemisslab.com/zh-Hant/questions/) · [原題批次](https://ocme.evemisslab.com/zh-Hant/practice/batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations/#q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0049) 本頁是已完成第二對話 AI 數理與中英語義覆核的教學詳解，只支持本題明示的模型與前提。沒有人工審查或新 Lean 證明，也不提升原題或 MKO 的 Canonical 真值狀態。 R0034-17524 · q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0049 ## 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":[23,24],"coefficients":[24,-25],"shift":2,"epsilon":[1,33],"point":1}。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":[23,24],"coefficients":[24,-25],"shift":2,"epsilon":[1,33],"point":1}。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](https://ocme.evemisslab.com/zh-Hant/mko/mko-pointwise-convergence-of-functions/) ## 逐步推導 S001 · implies操作前 same COMPLETE finite domain Fin(2);original aligned limits=[23, 24];signed coefficients=[24, -25];positive shift=2 操作後 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) 條件：無 完整共同有限domain與全尾公式是題目明示前提，不從樣本猜出。函數值與誤差不是機率，可帶負號或超過1；不截斷、不平移索引。 步驟識別與依賴規則：pointwise_explicit_common_domain_all_n_tail_not_probability · 依賴：G001 S002 · calculation操作前 selected pointwise witness absolute coefficient cross product := mul(25, 33) 操作後 selected pointwise witness absolute coefficient cross product = 825 條件：無 依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S001 S003 · calculation操作前 selected pointwise witness strict reciprocal threshold := div(825, 1) 操作後 selected pointwise witness strict reciprocal threshold = 825 條件：無 依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S002 S004 · calculation操作前 selected pointwise witness raw strict integer cutoff := sum(825, -2, 1) 操作後 selected pointwise witness raw strict integer cutoff = 824 條件：無 依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S003 S005 · implies操作前 selected pointwise witness:point=1;SIGNED coefficient=-25;absolute coefficient=25;positive epsilon=1/33;shift=2 操作後 threshold=825;floor=825;least nonnegative strict cutoff N=824 條件：無 |c|/(N+s)|c|b/a，故floor後加1再減shift，僅與0取max。等號不夠；零係數N=0。這裡沒有機率≤1的額外guard，不改原答案索引。 步驟識別與依賴規則：pointwise_least_strict_cutoff_floor_not_probability_guard · 依賴：G001, S004 S006 · calculation操作前 selected pointwise witness cutoff denominator := sum(824, 2) 操作後 selected pointwise witness cutoff denominator = 826 條件：無 依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S005 S007 · calculation操作前 selected pointwise witness absolute error at cutoff := div(25, 826) 操作後 selected pointwise witness absolute error at cutoff = 25/826 條件：無 依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S006 S008 · calculation操作前 selected pointwise witness previous denominator := sum(823, 2) 操作後 selected pointwise witness previous denominator = 825 條件：無 依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S007 S009 · calculation操作前 selected pointwise witness previous absolute error := div(25, 825) 操作後 selected pointwise witness previous absolute error = 1/33 條件：無 依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。 步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S008 S010 · implies操作前 selected pointwise witness:current N=824;current error=25/826;previous error=1/33;epsilon=1/33 操作後 current strict bound holds;previous fails;all smaller indices fail by monotonicity 條件：無 前一項仍≥epsilon，分母隨n增加，故所有更小非負n亦不合嚴格界。 步驟識別與依賴規則：pointwise_minimality_previous_index_strict_threshold · 依賴：G001, S009 S011 · implies操作前 selected pointwise witness:declared exact tail;positive denominators increase for all n>=N=824 操作後 for EVERY later n,absolute error<=bound=25/826