第 21433 題：實數列Bolzano－Weierstrass定理

原題與已知條件

whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[8,9],"coefficient":8,"shift":9,"parity":1,"epsilon":[1,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={"limit":[8,9],"coefficient":8,"shift":9,"parity":1,"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-bolzano-weierstrass-theorem-real

逐步推導

source operation=bw-oscillatory-tail;original snapshot and mathematics retained

explicit ordered source output fields=[parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate]

ORIGINAL a_n=8/9+(-1)^n*(8)/(n+9) for EVERY n>=0;parity=1

k(j)=2j+1 for ALL j>=0;k(j+1)-k(j)=2>0;selected sign=-1;whole sequence bounded by |L|+c/shift

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

bw c over epsilon = 136

bw index multiple := mul(2, 64)

bw index multiple = 128

bw actual source index := sum(1, 128)

bw actual source index = 129

bw witness denominator := sum(129, 9)

bw witness denominator = 138

bw witness distance := div(8, 138)

bw witness distance = 4/69

bw predecessor distance := div(8, 136)

bw predecessor distance = 1/17

bw:c=8,shift=9,epsilon=1/17;index(q)=2q+1;c/epsilon=136;floor=136

least nonnegative witness=64;actual source index=129;distance=4/69<epsilon;predecessor distance=1/17;no predecessor evaluated when witness0

bw:positive c=8,epsilon=1/17,shift=9,stride=2,offset=1;least witness=64

for EVERY integer q>=64,denominator=2q+1+9>=witness denominator>0,so 0<distance<=(4/69)<epsilon

bw at selected k(J) denominator := sum(129, 9)

bw at selected k(J) denominator = 138

bw at selected k(J) unsigned tail := div(8, 138)

bw at selected k(J) unsigned tail = 4/69

bw at selected k(J) signed value := sub(8/9, 4/69)

bw at selected k(J) signed value = 172/207

逐步反向核對

• S013 · premise_and_result_audit — 以精確有理數重算 bw at selected k(J) signed value，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 bw at selected k(J) unsigned tail，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 bw at selected k(J) denominator，不以小數近似替代。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S008 · premise_and_result_audit — 以精確有理數重算 bw predecessor distance，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 bw witness distance，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 bw witness denominator，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 bw actual source index，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 bw index multiple，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 bw c over epsilon，不以小數近似替代。
• 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

(1, k(j)=2j+1, 8/9, 1/17, 64, 172/207, 是, forall_j_ge_J_symbolic_subsequence_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。