{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-complete-metric-space"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-complete-metric-space-01","title":{"zh-Hant":"完備度量空間：以最小正距離證明有限度量空間完備，並以 (0,1] 對照","en":"Complete metric space: proving every finite metric space complete by the least positive distance, contrasted with (0,1]"},"summary":{"zh-Hant":"證明：每個至少有一點的有限度量空間 (X,d) 都是完備的（單點情形直接成立；至少兩點時用最小正距離 δ 證明每個 Cauchy 序列最終常值）。在 X0={0,1,3}、d(x,y)=|x−y| 上算出 δ=1，給一個最終常值證書與一個明示的非 Cauchy 序列；最後證明子空間 Y=(0,1] 不完備：x_n=1/n 在 Y 中是 Cauchy，但在 Y 中沒有極限。","en":"Prove that every finite metric space (X,d) with at least one point is complete (immediate for a single point; with at least two points, the least positive distance δ forces every Cauchy sequence to be ultimately constant). On X0={0,1,3} with d(x,y)=|x−y|, compute δ=1, give an ultimately constant certificate and an explicit non-Cauchy sequence; finally prove that the subspace Y=(0,1] is not complete: x_n=1/n is Cauchy in Y but has no limit in Y."},"mko_ids":["mko-complete-metric-space"]}],"title":"概念推導導讀"}