{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-complete-metric-space-01",
  "mko_id": "mko-complete-metric-space",
  "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]"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-complete-metric-space-01",
    "source": {
      "mko_id": "mko-complete-metric-space",
      "commit": "b28a5edbd46279b1db2a3f3f48bb3f447e18899f",
      "sha256": "783e45783b1dc07ccecb89f0c76b128f6c4a9a3da036e1678e908f2250530c73",
      "english_sha256": "505d9fc575b5de4a835e628a71c116b9f8b9518c9e79dc126f4b2e6f7d011b6d"
    },
    "mode": "theorem_proof",
    "audience_prerequisites": [
      "metric_def",
      "order_basics",
      "abs_rules",
      "archimedean",
      "min_finite"
    ],
    "goal": {
      "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."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "(X,d) 是有限度量空間，X 至少有一點，度量表完整。",
          "en": "(X,d) is a finite metric space, X has at least one point, and the metric table is complete."
        }
      },
      {
        "id": "G2",
        "statement": {
          "zh-Hant": "例子：X0={0,1,3}，d(x,y)=|x−y|（ℝ 的通常距離限制在 X0 上）。",
          "en": "Example: X0={0,1,3} with d(x,y)=|x−y| (the usual distance of ℝ restricted to X0)."
        }
      },
      {
        "id": "G3",
        "statement": {
          "zh-Hant": "對照：Y=(0,1]={t∈ℝ : 0<t≤1}，d(s,t)=|s−t|；序列 x_n=1/n（n≥1）。",
          "en": "Contrast: Y=(0,1]={t∈ℝ : 0<t≤1} with d(s,t)=|s−t|; the sequence x_n=1/n (n≥1)."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "mko_cauchy_def",
          "mko_conv_def"
        ],
        "before_tex": "(X,d)\\ \\text{complete}",
        "after_tex": "\\forall (x_n)\\subseteq X\\ \\text{Cauchy}\\ \\exists L\\in X:\\ x_n\\to L",
        "relation": "definition",
        "rule": "mko_complete_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "依 MKO，完備是空間的性質：X 中每一個 Cauchy 序列都必須收斂到 X 中的某點。Cauchy 的定義不預設極限存在；收斂要求對每個 ε>0 都有門檻 N。所以證明必須對任意 Cauchy 序列成立，不能只測幾條。",
          "en": "By the MKO, completeness is a property of the space: every Cauchy sequence in X must converge to some point of X. The Cauchy definition does not presuppose a limit; convergence requires a threshold N for every ε>0. So the proof must work for an arbitrary Cauchy sequence, not for a few tested ones."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "mko_complete_def",
            "mko_cauchy_def",
            "mko_conv_def"
          ],
          "explanation": {
            "zh-Hant": "此步只展開三個定義，不推出新事實。",
            "en": "This step only unfolds the three definitions and derives no new fact."
          }
        },
        "leaf_basis": "mko_complete_def + mko_cauchy_def + mko_conv_def + def_unfold"
      },
      {
        "id": "S02",
        "uses": [
          "G1"
        ],
        "before_tex": "|X|\\ge 1",
        "after_tex": "|X|=1\\ \\lor\\ |X|\\ge 2",
        "relation": "case_split",
        "rule": "case_analysis",
        "conditions": [],
        "explanation": {
          "zh-Hant": "X 至少有一點，所以恰有兩種情況：只有一點，或至少兩點。最小正距離只在至少兩點時有定義（MKO 假設 a1），所以分開處理。",
          "en": "X has at least one point, so exactly two cases arise: one point, or at least two. The least positive distance is defined only with at least two points (MKO assumption a1), so the cases are treated separately."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "case_analysis"
          ],
          "explanation": {
            "zh-Hant": "兩種情況涵蓋所有至少一點的有限集合，且互斥。",
            "en": "The two cases cover every finite set with at least one point and do not overlap."
          }
        },
        "leaf_basis": "case_analysis + order_basics"
      },
      {
        "id": "S03",
        "uses": [
          "S01",
          "S02",
          "metric_def"
        ],
        "before_tex": "X=\\{p\\}",
        "after_tex": "x_n=p\\ \\forall n,\\ d(x_n,p)=0<\\varepsilon\\ \\Rightarrow\\ x_n\\to p\\in X",
        "relation": "implies",
        "rule": "mko_conv_def",
        "conditions": [
          "|X|=1"
        ],
        "explanation": {
          "zh-Hant": "單點情形：X 中任何序列的每一項都是 p，d(x_n,p)=d(p,p)=0。對每個 ε>0 取 N=1，所有 n≥1 都有 0<ε，所以序列收斂到 p∈X。每個 Cauchy 序列（其實是每個序列）都收斂，X 完備。",
          "en": "Single point: every term of every sequence in X is p, so d(x_n,p)=d(p,p)=0. For each ε>0 take N=1; every n≥1 has 0<ε, so the sequence converges to p∈X. Every Cauchy sequence (indeed every sequence) converges, so X is complete."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02",
            "metric_def",
            "mko_conv_def"
          ],
          "explanation": {
            "zh-Hant": "只用 d(p,p)=0；這個情況不需要最小正距離。",
            "en": "It uses only d(p,p)=0; this case does not need the least positive distance."
          }
        },
        "leaf_basis": "mko_conv_def + metric_def + forall_intro"
      },
      {
        "id": "S04",
        "uses": [
          "S02",
          "G1",
          "min_finite"
        ],
        "before_tex": "D=\\{d(x,y): x,y\\in X,\\ x\\neq y\\}",
        "after_tex": "\\delta=\\min D",
        "relation": "definition",
        "rule": "def_intro",
        "conditions": [
          "|X|\\ge 2"
        ],
        "explanation": {
          "zh-Hant": "至少兩點時，D 非空（有一對相異點）且有限（度量表有限），所以最小值 δ 存在並屬於 D。這是 MKO 所說的最小正距離。",
          "en": "With at least two points, D is nonempty (some pair of distinct points exists) and finite (the metric table is finite), so its minimum δ exists and lies in D. This is the least positive distance of the MKO."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02",
            "min_finite"
          ],
          "explanation": {
            "zh-Hant": "需要有限與非空兩個條件；無限空間中這樣的最小值可能不存在（例如 Y 的距離可任意小）。",
            "en": "It needs both finiteness and nonemptiness; in an infinite space such a minimum may fail to exist (distances in Y can be arbitrarily small)."
          }
        },
        "leaf_basis": "def_intro + min_finite"
      },
      {
        "id": "S05",
        "uses": [
          "S04"
        ],
        "before_tex": "\\delta=\\min D",
        "after_tex": "\\delta>0,\\ d(x,y)<\\delta\\Rightarrow x=y",
        "relation": "theorem_application",
        "rule": "L_small_dist",
        "conditions": [
          "|X|\\ge 2"
        ],
        "explanation": {
          "zh-Hant": "依引理 L_small_dist（S05.1、S05.2 證明）：δ 是正數，而且距離小於 δ 的兩點必相同。這是整個證明的關鍵。",
          "en": "By Lemma L_small_dist (proved in S05.1 and S05.2): δ is positive, and two points at distance less than δ must coincide. This is the key to the whole proof."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S04",
            "L_small_dist"
          ],
          "explanation": {
            "zh-Hant": "引理的前提是有限且至少兩點；S04 已提供。",
            "en": "The lemma needs a finite space with at least two points, which S04 provides."
          }
        },
        "leaf_basis": "L_small_dist"
      },
      {
        "id": "S05.1",
        "uses": [
          "S04",
          "metric_def",
          "min_finite"
        ],
        "before_tex": "\\delta\\in D",
        "after_tex": "\\delta>0",
        "relation": "implies",
        "rule": "metric_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "δ 屬於 D，所以 δ=d(x0,y0) 對某對相異點 x0≠y0。度量的分離性給 d(x0,y0)≥0 且不為 0，所以 δ>0。",
          "en": "δ belongs to D, so δ=d(x0,y0) for some distinct x0≠y0. Separation of the metric gives d(x0,y0)≥0 and not 0, so δ>0."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "metric_def",
            "min_finite"
          ],
          "explanation": {
            "zh-Hant": "用到最小值被取到；若只是下確界而不被取到，可能是 0。此子步不使用 L_small_dist。",
            "en": "It uses that the minimum is attained; a mere infimum that is not attained could be 0. This sub-step does not use L_small_dist."
          }
        },
        "leaf_basis": "metric_def + min_finite + order_basics"
      },
      {
        "id": "S05.2",
        "uses": [
          "S04",
          "min_finite",
          "contrapositive"
        ],
        "before_tex": "x\\neq y\\Rightarrow d(x,y)\\in D\\Rightarrow d(x,y)\\ge\\delta",
        "after_tex": "d(x,y)<\\delta\\Rightarrow x=y",
        "relation": "equivalent",
        "rule": "contrapositive",
        "conditions": [],
        "explanation": {
          "zh-Hant": "若 x≠y，則 d(x,y) 是 D 的元素，不小於最小值 δ。取逆否：d(x,y)<δ 推出 x=y。S05.1 與 S05.2 合起來證得引理 L_small_dist。",
          "en": "If x≠y then d(x,y) is an element of D and is at least the minimum δ. Taking the contrapositive: d(x,y)<δ implies x=y. S05.1 and S05.2 together prove Lemma L_small_dist."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "contrapositive"
          ],
          "explanation": {
            "zh-Hant": "命題與其逆否等價，雙向成立。此子步不使用 L_small_dist。",
            "en": "A statement and its contrapositive are equivalent, so the step is two-way. This sub-step does not use L_small_dist."
          }
        },
        "leaf_basis": "contrapositive + min_finite + order_basics"
      },
      {
        "id": "S06",
        "uses": [
          "S01",
          "S05",
          "mko_cauchy_def"
        ],
        "before_tex": "(x_n)\\ \\text{Cauchy in}\\ X,\\ \\varepsilon=\\delta>0",
        "after_tex": "\\exists N\\ \\forall m,n\\ge N:\\ d(x_m,x_n)<\\delta",
        "relation": "implies",
        "rule": "forall_elim",
        "conditions": [
          "\\delta>0"
        ],
        "explanation": {
          "zh-Hant": "取 X 中任意一個 Cauchy 序列。Cauchy 的定義對每個 ε>0 都成立，特別是對 ε=δ（S05 保證 δ>0）：存在 N，使所有 m,n≥N 都有 d(x_m,x_n)<δ。",
          "en": "Take an arbitrary Cauchy sequence in X. The Cauchy definition holds for every ε>0, in particular for ε=δ (positive by S05): there is N with d(x_m,x_n)<δ for all m,n≥N."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "mko_cauchy_def",
            "S05",
            "forall_elim"
          ],
          "explanation": {
            "zh-Hant": "全稱消去需要 δ 在 ε 的範圍內，即 δ>0；不宣稱反向。",
            "en": "∀-elimination needs δ in the range of ε, that is δ>0; no converse is claimed."
          }
        },
        "leaf_basis": "forall_elim + mko_cauchy_def"
      },
      {
        "id": "S07",
        "uses": [
          "S05",
          "S06"
        ],
        "before_tex": "d(x_N,x_n)<\\delta\\ \\ (n\\ge N)",
        "after_tex": "x_n=x_N\\ \\ \\text{for all}\\ n\\ge N",
        "relation": "implies",
        "rule": "L_small_dist",
        "conditions": [],
        "explanation": {
          "zh-Hant": "在 S06 中取 m=N：對所有 n≥N，d(x_N,x_n)<δ，依引理 x_n=x_N。所以序列從第 N 項起恆等於 x_N：Cauchy 尾段最終常值。",
          "en": "Take m=N in S06: for all n≥N, d(x_N,x_n)<δ, so x_n=x_N by the lemma. The sequence is constant equal to x_N from index N on: the Cauchy tail is ultimately constant."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S05",
            "S06"
          ],
          "explanation": {
            "zh-Hant": "需要 m=N 在 m≥N 的範圍內；結論只關於 n≥N 的項，前面的項可以任意。",
            "en": "It needs m=N to lie in the range m≥N; the conclusion concerns only terms with n≥N, and earlier terms may be anything."
          }
        },
        "leaf_basis": "L_small_dist + forall_elim"
      },
      {
        "id": "S08",
        "uses": [
          "S07",
          "metric_def",
          "mko_conv_def"
        ],
        "before_tex": "L=x_N\\in X,\\ \\varepsilon>0",
        "after_tex": "\\forall n\\ge N:\\ d(x_n,L)=0<\\varepsilon\\ \\Rightarrow\\ x_n\\to L",
        "relation": "implies",
        "rule": "mko_conv_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "令 L=x_N，它是 X 的點。對任意 ε>0，用同一個 N：n≥N 時 d(x_n,L)=d(x_N,x_N)=0<ε。所以 x_n→L，且極限在 X 中。這就是 MKO 所說的「明示最終常值」完整證書。",
          "en": "Let L=x_N, a point of X. For any ε>0 use the same N: for n≥N, d(x_n,L)=d(x_N,x_N)=0<ε. So x_n→L, with the limit in X. This is the complete 'explicitly ultimately constant' certificate of the MKO."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S07",
            "metric_def",
            "mko_conv_def"
          ],
          "explanation": {
            "zh-Hant": "同一個 N 對所有 ε 都可用，因為尾段距離恰為 0；L∈X 因為它是序列的一項。",
            "en": "One N serves every ε because the tail distances are exactly 0; L∈X because it is a term of the sequence."
          }
        },
        "leaf_basis": "mko_conv_def + metric_def + forall_intro"
      },
      {
        "id": "S09",
        "uses": [
          "S06",
          "S07",
          "S08"
        ],
        "before_tex": "(x_n)\\ \\text{an arbitrary Cauchy sequence in}\\ X",
        "after_tex": "\\text{every Cauchy sequence in}\\ X\\ \\text{converges in}\\ X\\ \\ (|X|\\ge 2)",
        "relation": "implies",
        "rule": "forall_intro",
        "conditions": [
          "|X|\\ge 2"
        ],
        "explanation": {
          "zh-Hant": "S06 到 S08 對任意 Cauchy 序列成立，沒有對它加任何額外假設，所以依全稱推廣，至少兩點的有限度量空間中每個 Cauchy 序列都在 X 中收斂。",
          "en": "S06 to S08 hold for an arbitrary Cauchy sequence with no extra assumption on it, so by generalization every Cauchy sequence in a finite metric space with at least two points converges in X."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S06",
            "S07",
            "S08",
            "forall_intro"
          ],
          "explanation": {
            "zh-Hant": "全稱推廣的條件是序列為任意取；這裡沒有使用任何特定序列。",
            "en": "Generalization requires the sequence to be arbitrary; no particular sequence was used."
          }
        },
        "leaf_basis": "forall_intro"
      },
      {
        "id": "S10",
        "uses": [
          "S01",
          "S02",
          "S03",
          "S09"
        ],
        "before_tex": "|X|=1\\ \\lor\\ |X|\\ge 2",
        "after_tex": "(X,d)\\ \\text{complete}",
        "relation": "implies",
        "rule": "case_analysis",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩種情況都證得每個 Cauchy 序列在 X 中收斂（S03、S09），依 S01 的定義，每個至少一點的有限度量空間都完備。",
          "en": "In both cases every Cauchy sequence converges in X (S03, S09), so by the definition in S01 every finite metric space with at least one point is complete."
        },
        "reverse_audit": {
          "kind": "sufficient_goal_reduction",
          "required_ids": [
            "S01",
            "S02",
            "S03",
            "S09",
            "case_analysis"
          ],
          "explanation": {
            "zh-Hant": "目標 G：X 完備。足夠條件 H：兩種情況下每個 Cauchy 序列都收斂於 X；H⇒G 由定義與分情況論證。H 由 S03、S09 正向證成。",
            "en": "Goal G: X is complete. Sufficient condition H: in both cases every Cauchy sequence converges in X; H⇒G by the definition and proof by cases. H is proved forwards in S03 and S09."
          }
        },
        "leaf_basis": "case_analysis + mko_complete_def"
      },
      {
        "id": "S11",
        "uses": [
          "G2",
          "S04"
        ],
        "before_tex": "d(0,1),\\ d(0,3),\\ d(1,3)",
        "after_tex": "1,\\ 3,\\ 2;\\ \\delta=\\min(1,3,2)=1",
        "relation": "implies",
        "rule": "numeric_evaluation",
        "conditions": [],
        "explanation": {
          "zh-Hant": "X0 的三對相異點距離為 |0−1|=1、|0−3|=3、|1−3|=2，所以 D={1,2,3}，δ=1。d 是 ℝ 的通常距離限制在 X0 上，所以是度量；X0 由 S10 完備。",
          "en": "The three pairs of distinct points of X0 have distances |0−1|=1, |0−3|=3, |1−3|=2, so D={1,2,3} and δ=1. d is the usual distance of ℝ restricted to X0, hence a metric, and X0 is complete by S10."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G2",
            "numeric_evaluation"
          ],
          "explanation": {
            "zh-Hant": "完整距離表的三對都計算了；漏掉一對可能算錯 δ。",
            "en": "All three pairs of the complete distance table are computed; omitting one could give a wrong δ."
          }
        },
        "leaf_basis": "numeric_evaluation + abs_rules + min_finite"
      },
      {
        "id": "S12",
        "uses": [
          "S11",
          "mko_cauchy_def",
          "mko_conv_def"
        ],
        "before_tex": "x_1=0,\\ x_2=3,\\ x_n=1\\ (n\\ge 3)",
        "after_tex": "(x_n)\\ \\text{Cauchy with}\\ N=3,\\ x_n\\to 1\\in X_0",
        "relation": "implies",
        "rule": "mko_conv_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "這是一個明示的最終常值證書：序列宣告第 3 項以後全部等於 1。對每個 ε>0 取 N=3：m,n≥3 時 d(x_m,x_n)=0<ε（Cauchy），d(x_n,1)=0<ε（收斂到 1）。前兩項不影響結論。",
          "en": "This is an explicit ultimately constant certificate: the sequence declares that every term from the 3rd on equals 1. For each ε>0 take N=3: for m,n≥3, d(x_m,x_n)=0<ε (Cauchy) and d(x_n,1)=0<ε (convergence to 1). The first two terms do not affect the conclusion."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "mko_cauchy_def",
            "mko_conv_def"
          ],
          "explanation": {
            "zh-Hant": "證書依賴「之後所有項都等於 1」這個明示宣告，而不是列出的有限前綴。",
            "en": "The certificate rests on the explicit declaration that all later terms equal 1, not on a listed finite prefix."
          }
        },
        "leaf_basis": "mko_conv_def + mko_cauchy_def + forall_intro"
      },
      {
        "id": "S13",
        "uses": [
          "G2",
          "mko_cauchy_def"
        ],
        "before_tex": "x_n=0\\ (n\\ \\text{odd}),\\ x_n=1\\ (n\\ \\text{even})",
        "after_tex": "\\text{not Cauchy: for}\\ \\varepsilon=\\tfrac12,\\ d(x_N,x_{N+1})=1\\ge\\tfrac12\\ \\forall N",
        "relation": "implies",
        "rule": "quantifier_negation",
        "conditions": [],
        "explanation": {
          "zh-Hant": "交替序列 0、1、0、1 等：對 ε=1/2，任給 N，取 m=N、n=N+1，兩項一個是 0、一個是 1，距離 1≥1/2。所以「存在 N 使所有 m,n≥N 距離小於 1/2」為假，序列不是 Cauchy。它也不收斂，但這不否證完備性，因為完備只要求 Cauchy 序列收斂。",
          "en": "The alternating sequence 0, 1, 0, 1 and so on: for ε=1/2 and any N, take m=N and n=N+1; one term is 0 and the other 1, at distance 1≥1/2. So 'some N makes every distance with m,n≥N less than 1/2' is false, and the sequence is not Cauchy. It does not converge either, but that does not refute completeness, which only asks Cauchy sequences to converge."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "mko_cauchy_def",
            "quantifier_negation"
          ],
          "explanation": {
            "zh-Hant": "否定「∀ε∃N∀m,n」只需一個 ε 與對每個 N 的一組反例；這是明示的非 Cauchy 反例。",
            "en": "Negating '∀ε ∃N ∀m,n' needs one ε and, for each N, one failing pair; this is an explicit non-Cauchy counterexample."
          }
        },
        "leaf_basis": "quantifier_negation + mko_cauchy_def"
      },
      {
        "id": "S14",
        "uses": [
          "G3",
          "mko_cauchy_def",
          "archimedean"
        ],
        "before_tex": "x_n=\\tfrac1n\\in Y,\\ \\varepsilon>0",
        "after_tex": "\\exists N:\\ \\forall m,n\\ge N,\\ \\left|\\tfrac1m-\\tfrac1n\\right|<\\tfrac1N<\\varepsilon",
        "relation": "implies",
        "rule": "archimedean",
        "conditions": [],
        "explanation": {
          "zh-Hant": "每個 1/n 都在 (0,1]，所以序列在 Y 中。給 ε>0，依阿基米德性質取 N 使 1/N<ε。若 m,n≥N，則 1/m 與 1/n 都在 (0,1/N]；兩個正數之差的絕對值小於較大者，所以 |1/m−1/n|<max(1/m,1/n)≤1/N<ε。故 (x_n) 在 Y 中是 Cauchy。",
          "en": "Each 1/n lies in (0,1], so the sequence lies in Y. Given ε>0, the Archimedean property gives N with 1/N<ε. If m,n≥N, then 1/m and 1/n both lie in (0,1/N]; the difference of two positive numbers is smaller in absolute value than the larger one, so |1/m−1/n|<max(1/m,1/n)≤1/N<ε. Hence (x_n) is Cauchy in Y."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "archimedean",
            "mko_cauchy_def",
            "abs_rules"
          ],
          "explanation": {
            "zh-Hant": "用到「兩正數之差小於較大者」與阿基米德性質；這是對所有 ε 的倒數序列界證書，N6 只是有限範圍內的核對。",
            "en": "It uses 'the difference of two positive numbers is less than the larger' and the Archimedean property; this is a reciprocal-bound certificate for every ε, and N6 is only a check on a finite range."
          }
        },
        "leaf_basis": "archimedean + abs_rules + order_basics + forall_intro"
      },
      {
        "id": "S15",
        "uses": [
          "G3",
          "archimedean",
          "mko_conv_def"
        ],
        "before_tex": "\\left|\\tfrac1n-0\\right|=\\tfrac1n",
        "after_tex": "x_n\\to 0\\ \\text{in}\\ \\mathbb{R}",
        "relation": "implies",
        "rule": "mko_conv_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "在 ℝ 中：給 ε>0，取 N 使 1/N<ε，則 n≥N 時 |1/n−0|=1/n≤1/N<ε。所以 x_n 在 ℝ 中收斂到 0，而 0∉Y。",
          "en": "In ℝ: given ε>0, take N with 1/N<ε; then for n≥N, |1/n−0|=1/n≤1/N<ε. So x_n converges to 0 in ℝ, and 0∉Y."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "archimedean",
            "mko_conv_def"
          ],
          "explanation": {
            "zh-Hant": "這是在外圍空間 ℝ 中的收斂；它本身不說明 Y 是否完備。",
            "en": "This is convergence in the ambient space ℝ; on its own it says nothing about whether Y is complete."
          }
        },
        "leaf_basis": "mko_conv_def + archimedean + abs_rules"
      },
      {
        "id": "S16",
        "uses": [
          "metric_def"
        ],
        "before_tex": "x_n\\to L,\\ x_n\\to L'",
        "after_tex": "L=L'",
        "relation": "theorem_application",
        "rule": "L_limit_unique",
        "conditions": [],
        "explanation": {
          "zh-Hant": "引理 L_limit_unique（S16.1、S16.2 證明）：度量空間中的極限唯一。下一步需要它把 Y 中的極限與 ℝ 中的極限 0 連起來。",
          "en": "Lemma L_limit_unique (proved in S16.1 and S16.2): limits in a metric space are unique. The next step needs it to connect a limit in Y with the limit 0 in ℝ."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "metric_def",
            "L_limit_unique"
          ],
          "explanation": {
            "zh-Hant": "只需度量公理；對任何度量空間成立。",
            "en": "It needs only the metric axioms and holds in every metric space."
          }
        },
        "leaf_basis": "L_limit_unique"
      },
      {
        "id": "S16.1",
        "uses": [
          "metric_def",
          "mko_conv_def"
        ],
        "before_tex": "d(x_n,L)<\\varepsilon\\ (n\\ge N_1),\\ d(x_n,L')<\\varepsilon\\ (n\\ge N_2)",
        "after_tex": "d(L,L')<2\\varepsilon\\ \\ \\text{for every}\\ \\varepsilon>0",
        "relation": "implies",
        "rule": "metric_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "給 ε>0，由兩個收斂各得 N1、N2；取 n=max(N1,N2)。依對稱與三角不等式，d(L,L′)≤d(L,x_n)+d(x_n,L′)<ε+ε=2ε。",
          "en": "Given ε>0, the two convergences give N1 and N2; take n=max(N1,N2). By symmetry and the triangle inequality, d(L,L′)≤d(L,x_n)+d(x_n,L′)<ε+ε=2ε."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "metric_def",
            "mko_conv_def"
          ],
          "explanation": {
            "zh-Hant": "需要一個同時大於兩個門檻的 n；此子步不使用 L_limit_unique。",
            "en": "It needs one n beyond both thresholds; this sub-step does not use L_limit_unique."
          }
        },
        "leaf_basis": "metric_def + mko_conv_def + order_basics"
      },
      {
        "id": "S16.2",
        "uses": [
          "S16.1",
          "metric_def",
          "contradiction"
        ],
        "before_tex": "d(L,L')<2\\varepsilon\\ \\forall\\varepsilon>0",
        "after_tex": "L=L'",
        "relation": "implies",
        "rule": "contradiction",
        "conditions": [],
        "explanation": {
          "zh-Hant": "若 L≠L′，則 r=d(L,L′)>0；取 ε=r/2 得 r<r，矛盾。所以 L=L′。S16.1 與 S16.2 合起來證得引理 L_limit_unique。",
          "en": "If L≠L′ then r=d(L,L′)>0; taking ε=r/2 gives r<r, a contradiction. So L=L′. S16.1 and S16.2 together prove Lemma L_limit_unique."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S16.1",
            "metric_def",
            "contradiction"
          ],
          "explanation": {
            "zh-Hant": "用到分離性：相異點距離為正；此子步不使用 L_limit_unique。",
            "en": "It uses separation: distinct points are at positive distance; this sub-step does not use L_limit_unique."
          }
        },
        "leaf_basis": "contradiction + metric_def + order_basics"
      },
      {
        "id": "S17",
        "uses": [
          "S14",
          "S15",
          "S16",
          "G3"
        ],
        "before_tex": "x_n\\to L\\ \\text{in}\\ Y\\ \\text{for some}\\ L\\in Y",
        "after_tex": "L=0\\notin Y\\ \\Rightarrow\\ Y\\ \\text{not complete}",
        "relation": "implies",
        "rule": "contradiction",
        "conditions": [],
        "explanation": {
          "zh-Hant": "假設 x_n 在 Y 中收斂到某個 L∈Y。Y 的距離就是 ℝ 的距離，所以 x_n 在 ℝ 中也收斂到 L；由 S15 與引理 L_limit_unique，L=0，但 0∉Y，矛盾。故 Cauchy 序列 (x_n)（S14）在 Y 中沒有極限，Y 不完備。序列在外圍空間 ℝ 有極限並不使子空間完備。",
          "en": "Suppose x_n converges in Y to some L∈Y. The distance of Y is that of ℝ, so x_n also converges to L in ℝ; by S15 and Lemma L_limit_unique, L=0, but 0∉Y, a contradiction. So the Cauchy sequence (x_n) of S14 has no limit in Y, and Y is not complete. Having a limit in the ambient space ℝ does not make the subspace complete."
        },
        "reverse_audit": {
          "kind": "sufficient_goal_reduction",
          "required_ids": [
            "S14",
            "S15",
            "S16",
            "contradiction"
          ],
          "explanation": {
            "zh-Hant": "目標 G：Y 不完備。足夠條件 H：存在 Y 中的 Cauchy 序列在 Y 中不收斂；H⇒G 由完備的定義（否定全稱只需一個反例）。H 由 S14 到 S16 正向證成。",
            "en": "Goal G: Y is not complete. Sufficient condition H: some Cauchy sequence in Y does not converge in Y; H⇒G by the definition of completeness (one counterexample refutes a universal). H is proved forwards in S14 to S16."
          }
        },
        "leaf_basis": "contradiction + quantifier_negation + L_limit_unique"
      }
    ],
    "result": {
      "zh-Hant": "每個至少一點的有限度量空間都完備：單點時每個序列恆等；至少兩點時取最小正距離 δ>0，Cauchy 定義用 ε=δ 迫使序列從某個 N 起恆等於 x_N，因而收斂到 x_N∈X。X0={0,1,3} 的 δ=1；序列 0、3、1、1、1 等以 N=3 為最終常值證書；交替的 0、1 不是 Cauchy。子空間 (0,1] 不完備：1/n 是 Cauchy，但唯一可能的極限 0 不在 (0,1] 中。",
      "en": "Every finite metric space with at least one point is complete: with one point every sequence is constant; with at least two, the least positive distance δ>0 used as ε in the Cauchy definition forces the sequence to equal x_N from some N on, so it converges to x_N∈X. For X0={0,1,3}, δ=1; the sequence 0, 3, 1, 1, 1 and so on has the ultimately constant certificate N=3; the alternating 0, 1 sequence is not Cauchy. The subspace (0,1] is not complete: 1/n is Cauchy, but its only possible limit 0 is not in (0,1]."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "(0,1] 完備，因為 1/n 在 ℝ 中有極限 0",
          "en": "(0,1] is complete because 1/n has the limit 0 in ℝ"
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "0∉(0,1]，而極限唯一，所以 1/n 在 (0,1] 中沒有極限。",
          "en": "0∉(0,1] and limits are unique, so 1/n has no limit in (0,1]."
        }
      },
      {
        "claim": {
          "zh-Hant": "X0 中交替序列 0、1、0、1 等不收斂，所以 X0 不完備",
          "en": "The alternating sequence 0, 1, 0, 1 and so on in X0 does not converge, so X0 is not complete"
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "它不是 Cauchy（S13）；完備只要求 Cauchy 序列收斂。",
          "en": "It is not Cauchy (S13); completeness only requires Cauchy sequences to converge."
        }
      },
      {
        "claim": {
          "zh-Hant": "在 (0,1] 中測了五條 Cauchy 序列都收斂於 (0,1]，所以 (0,1] 完備",
          "en": "Five Cauchy sequences in (0,1] were tested and all converged in (0,1], so (0,1] is complete"
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "1/n 是 (0,1] 中的 Cauchy 序列，在 (0,1] 中沒有極限。",
          "en": "1/n is a Cauchy sequence in (0,1] with no limit in (0,1]."
        }
      },
      {
        "claim": {
          "zh-Hant": "有限度量空間中的 Cauchy 序列從第一項起就是常數",
          "en": "In a finite metric space a Cauchy sequence is constant from its first term"
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "序列 0、3、1、1、1 等在 X0 中是 Cauchy，但前兩項不同；只能保證最終常值。",
          "en": "The sequence 0, 3, 1, 1, 1 and so on is Cauchy in X0 but its first two terms differ; only ultimately constant is guaranteed."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "metric_def",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "度量：d:X×X→ℝ 滿足 d(x,y)≥0 且 d(x,y)=0 ⇔ x=y；d(x,y)=d(y,x)；三角不等式 d(x,z)≤d(x,y)+d(y,z)。",
          "en": "Metric: d:X×X→ℝ with d(x,y)≥0 and d(x,y)=0 ⇔ x=y; d(x,y)=d(y,x); triangle inequality d(x,z)≤d(x,y)+d(y,z)."
        }
      },
      {
        "id": "order_basics",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "實數的序：任兩數恰有 u<v、u=v、u>v 之一；u<v 且 v<w 推出 u<w；u<v 推出 u+w<v+w；u<v 且 w>0 推出 uw<vw；u<v 且 w<0 推出 uw>vw。",
          "en": "Order of the reals: exactly one of u<v, u=v, u>v holds; u<v and v<w give u<w; u<v gives u+w<v+w; u<v and w>0 give uw<vw; u<v and w<0 give uw>vw."
        }
      },
      {
        "id": "abs_rules",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "絕對值（實數）：|u|=u（u≥0）、|u|=−u（u<0）；|u|≥0；|uv|=|u||v|；三角不等式 |u+v|≤|u|+|v|；|u|<r ⇔ −r<u<r。",
          "en": "Absolute value (reals): |u|=u if u≥0 and |u|=−u if u<0; |u|≥0; |uv|=|u||v|; triangle inequality |u+v|≤|u|+|v|; |u|<r ⇔ −r<u<r."
        }
      },
      {
        "id": "archimedean",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "阿基米德性質：對任意實數 x，存在自然數 n>x；因此對任意 ε>0，存在 n 使 1/n<ε。",
          "en": "Archimedean property: for every real x there is a natural number n>x; hence for every ε>0 there is n with 1/n<ε."
        }
      },
      {
        "id": "def_unfold",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "展開定義：把一個已定義的詞換成它的定義內容（或反向收回），命題的真假不變。",
          "en": "Unfolding a definition: replacing a defined term by its definition (or folding it back) does not change whether a statement is true."
        }
      },
      {
        "id": "case_analysis",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "分情況論證：若 P∨Q 成立，且 P⇒R、Q⇒R 都已證，則 R 成立。所有情況必須涵蓋全部可能。",
          "en": "Proof by cases: if P∨Q holds and both P⇒R and Q⇒R are proved, then R holds. The cases must cover every possibility."
        }
      },
      {
        "id": "forall_intro",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "∀引入（全稱推廣）：若某命題對一個「任意的」元素 x 證成，而證明沒有對 x 加任何額外假設，則它對論域中所有 x 成立。",
          "en": "∀-introduction (generalization): if a statement is proved for an arbitrary element x, with no extra assumption on x, it holds for every x in the domain."
        }
      },
      {
        "id": "def_intro",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "引入定義：可為一個命題、式子或物件取名；之後名稱與被定義者可互相替換。",
          "en": "Introducing a definition: a statement, expression or object may be given a name, after which the name and what it stands for are interchangeable."
        }
      },
      {
        "id": "contrapositive",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "逆否命題：P⇒Q 與 ¬Q⇒¬P 等價。逆命題 Q⇒P 一般不與 P⇒Q 等價。",
          "en": "Contrapositive: P⇒Q is equivalent to ¬Q⇒¬P. The converse Q⇒P is in general not equivalent to P⇒Q."
        }
      },
      {
        "id": "forall_elim",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "∀消去（代入特例）：若 ∀x∈D, P(x) 成立且 a∈D，則 P(a) 成立。",
          "en": "∀-elimination (instantiation): if ∀x∈D, P(x) holds and a∈D, then P(a) holds."
        }
      },
      {
        "id": "numeric_evaluation",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "明確整數或有理數算式的精確求值；可以獨立重算。",
          "en": "Exact evaluation of an explicit integer or rational expression; independently recomputable."
        }
      },
      {
        "id": "quantifier_negation",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "量詞否定：¬(∀x, P(x)) 等價於 ∃x, ¬P(x)；¬(∃x, P(x)) 等價於 ∀x, ¬P(x)。",
          "en": "Negating quantifiers: ¬(∀x, P(x)) is equivalent to ∃x, ¬P(x), and ¬(∃x, P(x)) is equivalent to ∀x, ¬P(x)."
        }
      },
      {
        "id": "contradiction",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "反證法：若假設 ¬P 推出矛盾（某命題與其否定同時成立），則 P 成立。",
          "en": "Proof by contradiction: if assuming ¬P leads to a contradiction (a statement and its negation both hold), then P holds."
        }
      },
      {
        "id": "mko_complete_def",
        "kind": "cited",
        "source": {
          "mko_id": "mko-complete-metric-space",
          "sha256": "783e45783b1dc07ccecb89f0c76b128f6c4a9a3da036e1678e908f2250530c73",
          "commit": "b28a5edbd46279b1db2a3f3f48bb3f447e18899f"
        },
        "statement": {
          "zh-Hant": "獨立來源追加式勘誤（使用者已採用於本地修訂；未合併或發布）：度量空間 (X,d) 完備，若 X 內每個 Cauchy 序列都收斂到 X 內某點。有限空間至少有兩點時，有限非空的正距離集合有最小值，迫使 Cauchy 尾段最終常值；單點情形所有序列本已常值，不需未定義的最小正距離。抽樣若干序列不足以證一般完備性。原始引文與 canonical MKO 保留。",
          "en": "Independent append-only source erratum (owner-adopted in this local revision; not merged or published): a metric space (X,d) is complete if every Cauchy sequence in X converges to a point of X. For a finite space with at least two points, the finite nonempty positive-distance set has a minimum forcing Cauchy tails to be ultimately constant. In the singleton case every sequence is already constant, with no minimum positive distance required or defined. Sampled sequences do not prove general completeness. The original citation and canonical MKO are preserved."
        }
      },
      {
        "id": "mko_cauchy_def",
        "kind": "cited",
        "source": {
          "mko_id": "mko-cauchy-sequence",
          "sha256": "f833bc771b9e0a259993b3560086625a99bfce9069edf8b08acd37a09b08c004",
          "commit": "b28a5edbd46279b1db2a3f3f48bb3f447e18899f"
        },
        "statement": {
          "zh-Hant": "序列(x_n)為Cauchy，若對每個epsilon>0存在N，使所有m,n>=N都有d(x_m,x_n)<epsilon。定義不先假設極限存在或位於空間內。",
          "en": "A sequence (x_n) is Cauchy if for every epsilon>0 there exists N such that all m,n≥N satisfy d(x_m,x_n)<epsilon. The definition does not first assume that a limit exists or belongs to the space."
        }
      },
      {
        "id": "mko_conv_def",
        "kind": "cited",
        "source": {
          "mko_id": "mko-metric-convergence",
          "sha256": "185db7b6b678014d6791d0448e2eccc1efc643b288d9b95c2b7b91f3ea710969",
          "commit": "b28a5edbd46279b1db2a3f3f48bb3f447e18899f"
        },
        "statement": {
          "zh-Hant": "序列x_n在度量空間中收斂到x，若對每個epsilon>0存在N，使所有n≥N都有d(x_n,x)<epsilon。有限樣本尾段通過不足以證一般收斂；明示最終常值的無限序列則可給完整witness。",
          "en": "A sequence x_n in a metric space converges to x if, for every epsilon>0, there exists N such that all n≥N satisfy d(x_n,x)<epsilon. Passing a sampled finite tail is insufficient for general convergence; an explicitly ultimately constant infinite sequence can instead provide a complete witness."
        }
      },
      {
        "id": "min_finite",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "非空有限實數集合有最小元素，而且最小元素屬於該集合。",
          "en": "A nonempty finite set of reals has a least element, and that least element belongs to the set."
        }
      },
      {
        "id": "L_small_dist",
        "kind": "lemma",
        "proved_in": [
          "S05.1",
          "S05.2"
        ],
        "statement": {
          "zh-Hant": "設有限度量空間至少有兩點，δ=min{d(x,y) : x≠y}。則 δ>0，且 d(x,y)<δ 推出 x=y。",
          "en": "In a finite metric space with at least two points let δ=min{d(x,y) : x≠y}. Then δ>0, and d(x,y)<δ implies x=y."
        }
      },
      {
        "id": "L_limit_unique",
        "kind": "lemma",
        "proved_in": [
          "S16.1",
          "S16.2"
        ],
        "statement": {
          "zh-Hant": "在度量空間中，若 x_n→L 且 x_n→L′，則 L=L′。",
          "en": "In a metric space, if x_n→L and x_n→L′, then L=L′."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N1",
        "step": "S11",
        "lhs": "abs(0-1)",
        "rhs": "1"
      },
      {
        "id": "N2",
        "step": "S11",
        "lhs": "abs(0-3)",
        "rhs": "3"
      },
      {
        "id": "N3",
        "step": "S11",
        "lhs": "abs(1-3)",
        "rhs": "2"
      },
      {
        "id": "N4",
        "step": "S11",
        "lhs": "min(abs(0-1),abs(0-3),abs(1-3))",
        "rhs": "1"
      },
      {
        "id": "N5",
        "step": "S13",
        "lhs": "abs(0-1)",
        "relation": ">=",
        "rhs": "1/2"
      },
      {
        "id": "N6",
        "step": "S14",
        "vars": [
          "m",
          "n"
        ],
        "range": [
          1,
          30
        ],
        "lhs": "abs(1/m-1/n)",
        "relation": "<",
        "rhs": "1/min(m,n)"
      },
      {
        "id": "N7",
        "step": "S15",
        "lhs": "1/101",
        "relation": "<",
        "rhs": "1/100"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "只在 MKO 的範圍內：有限證書用完整有限度量表且至少一點；最小正距離論證用至少兩點（假設 a1）；序列題只接受明示的最終常值證書或明示的非 Cauchy 反例，有限前綴不冒充完備證明（假設 a2）。(0,1] 的對照用倒數序列的界作完整證書。不處理一般完備化或 Banach 空間；不改動 canonical MKO 或英文 edition。",
      "en": "Stays inside the MKO: the finite certificate uses a complete finite metric table with at least one point; the least-positive-distance argument uses at least two (assumption a1); sequence exercises accept only an explicit ultimately constant certificate or an explicit non-Cauchy counterexample, and a finite prefix is not a completeness proof (assumption a2). The contrast with (0,1] uses reciprocal-sequence bounds as a complete certificate. Completions in general and Banach spaces are not treated; the canonical MKO and its English edition are not changed."
    },
    "md_symbols": [
      [
        "(X,d)",
        "有限度量空間",
        "finite metric space"
      ],
      [
        "(x_n)",
        "X 中的序列，n=1,2,3 等",
        "a sequence in X, n=1, 2, 3 and so on"
      ],
      [
        "\\delta",
        "最小正距離 min{d(x,y): x≠y}",
        "least positive distance min{d(x,y): x≠y}"
      ],
      [
        "N",
        "Cauchy 或收斂定義中的指標門檻",
        "index threshold in the Cauchy or convergence definition"
      ],
      [
        "X_0",
        "例子 {0,1,3}，d(x,y)=|x−y|",
        "the example {0,1,3} with d(x,y)=|x−y|"
      ],
      [
        "Y",
        "子空間 (0,1]，d(s,t)=|s−t|",
        "the subspace (0,1] with d(s,t)=|s−t|"
      ]
    ],
    "md_error_controls": [
      {
        "title": "完備是空間性質，不是測幾條序列 / Completeness is a property of the space, not of a few tested sequences",
        "zh": "MKO 指出的常見誤解是測過幾條序列都收斂就宣稱完備。在 Y=(0,1] 中，常數序列 1/2 與序列 1/2+1/(2n) 等都收斂於 Y，但 1/n 不收斂於 Y（S17）。完備必須對每一個 Cauchy 序列成立；有限空間的證明（S06 到 S09）就是對任意 Cauchy 序列進行的。",
        "en": "The common mistake named by the MKO is to declare completeness after a few tested sequences converge. In Y=(0,1], the constant sequence 1/2 and the sequence 1/2+1/(2n) both converge in Y, but 1/n does not (S17). Completeness must hold for every Cauchy sequence; the finite-space proof (S06 to S09) works with an arbitrary Cauchy sequence."
      },
      {
        "title": "外圍空間的極限不算數 / A limit in the ambient space does not count",
        "zh": "1/n 在 ℝ 中收斂到 0，但 0∉(0,1]，所以 (0,1] 不完備。反過來，X0={0,1,3} 完備是由 S10 的有限論證得到，而不是因為 ℝ 完備。完備性要求極限屬於這個空間本身。",
        "en": "1/n converges to 0 in ℝ, but 0∉(0,1], so (0,1] is not complete. Conversely, X0={0,1,3} is complete by the finite argument of S10, not because ℝ is complete. Completeness requires the limit to belong to the space itself."
      },
      {
        "title": "有限前綴不是證書 / A finite prefix is not a certificate",
        "zh": "列出一個序列的前 1000 項都等於 1，並不說明之後的項也是 1。MKO 只接受明示「之後所有項都等於尾值」的最終常值證書（S12）或明示的非 Cauchy 反例（S13）。同理，δ 必須由完整距離表的每一對求出（S11）。",
        "en": "Listing the first 1000 terms of a sequence as 1 does not show that later terms are 1. The MKO accepts only an ultimately constant certificate that explicitly declares every later term equal to the tail value (S12), or an explicit non-Cauchy counterexample (S13). Likewise δ must come from every pair of the complete distance table (S11)."
      }
    ],
    "md_open_items": [
      {
        "zh-Hant": "N6 只在 1≤m,n≤30 的有限範圍核對 |1/m−1/n|<1/min(m,n)；對所有 m,n 的證明是 S14 的論證。",
        "en": "N6 checks |1/m−1/n|<1/min(m,n) only on the finite range 1≤m,n≤30; the proof for all m, n is the argument of S14."
      },
      {
        "zh-Hant": "min_finite 是本稿的前提；錯誤控制中 1/2+1/(2n) 收斂於 Y 只作說明，未列為編號步驟。",
        "en": "min_finite is a premise of this guide; that 1/2+1/(2n) converges in Y is explanatory in the error controls and not a numbered step."
      }
    ]
  },
  "revision": {
    "id": "production-20261019-adopted-96",
    "source_commit": "b28a5edbd46279b1db2a3f3f48bb3f447e18899f",
    "adopted_source_sha256": "f777d86bce746681448ac94bc6f723103abd39b7d9520aa52006b6c641835c8d",
    "original_source_sha256": "1db5dbc4b9d2cc946ce1390b35ee86de434aefd9ce839d023e2612e22357f1e8",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "independent_AI_concurrence": "NOT_MEASURED",
    "issue_ids": [
      "BR-M072"
    ],
    "scope": "Owner-adopted natural-language teaching guide; not a formal proof, canonical MKO rewrite or evidence-status promotion."
  }
}