第 20569 題：上確界與下確界

原題與已知條件

非空且相應有界的實值集合或明示有界開區間。操作=supinf-open-interval-nonattainment；資料={"left":[18,19],"right":[20,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 非空且相應有界的實值集合或明示有界開區間。操作=supinf-open-interval-nonattainment；資料={"left":[18,19],"right":[20,19]}。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-supremum-and-infimum

逐步推導

source operation=supinf-open-interval-nonattainment;original snapshot and mathematics retained

explicit ordered source output fields=[left, right, supremum, infimum, supremum_attained, infimum_attained, boundary]

supinf WHOLE open interval positive width := sub(20/19, 18/19)

supinf WHOLE open interval positive width = 2/19

supinf interval midpoint numerator := sum(18/19, 20/19)

supinf interval midpoint numerator = 2

supinf one midpoint nonemptiness witness := div(2, 2)

supinf one midpoint nonemptiness witness = 1

WHOLE original real open interval(18/19,20/19);positive width=2/19;one displayed midpoint=1

for EVERY real x in this WHOLE interval,18/19<x<20/19;right=20/19 is an upper bound,left=18/19 is a lower bound;midpoint only witnesses nonemptiness

WHOLE interval left L=18/19,right R=20/19;consider ANY real proposed upper bound b<R or lower bound c>L

x=(max(L,b)+R)/2 lies in the interval and x>b,so b is NOT upper;y=(L+min(R,c))/2 lies in the interval and y<c,so c is NOT lower;therefore sup=R and inf=L

original interval conditions L<x<R with L=18/19,R=20/19

NEITHER endpoint satisfies both strict inequalities;supremum attained=否,infimum attained=否;for any interior point a midpoint toward each endpoint gives a larger/smaller interior point

逐步反向核對

• S007 · 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。
• S006 · 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。
• 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 supinf one midpoint nonemptiness witness，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 supinf interval midpoint numerator，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 supinf WHOLE open interval positive width，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

(18/19, 20/19, 20/19, 18/19, 否, 否, supremum_and_infimum_not_attained)

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。