{"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-metric-convergence"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-metric-convergence-01","title":{"zh-Hant":"度量空間中的收斂：有限度量表上最終常值序列的極限與唯一性","en":"Convergence in a metric space: limit and uniqueness for an ultimately constant sequence on a finite metric table"},"summary":{"zh-Hant":"在三點集合 X={a,b,c} 上給定完整度量表 d(a,b)=1、d(a,c)=2、d(b,c)=3/2。序列 x_0=a、x_1=c、x_2=a，並明示對所有 n≥3 有 x_n=b。(1) 核對 d 是度量；(2) 以 ε-N 定義證明 x_n→b，並找出各 ε 下最小的有效 N；(3) 證明度量空間中極限唯一，並直接給出 x_n 不收斂到 a、也不收斂到 c 的見證；(4) 用一個在有限尾段上與 x_n 完全一致、極限卻不同的序列，說明只核有限尾段不足以判定極限是否為指定的 b。","en":"On the three-point set X={a,b,c} a complete metric table is given: d(a,b)=1, d(a,c)=2, d(b,c)=3/2. The sequence has x_0=a, x_1=c, x_2=a, and it is declared that x_n=b for every n≥3. (1) Check that d is a metric; (2) prove x_n→b from the ε-N definition and find the least valid N for each ε; (3) prove that limits in a metric space are unique, and give direct witnesses that x_n converges neither to a nor to c; (4) using a sequence that agrees with x_n on a finite tail but has a different limit, show that checking a finite tail cannot decide whether the limit is the specified b."},"mko_ids":["mko-metric-convergence"]}],"title":"概念推導導讀"}