{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-monotone-sequence-convergence-theorem-01",
  "mko_id": "mko-monotone-sequence-convergence-theorem",
  "title": {
    "zh-Hant": "有界單調尾與上確界論證",
    "en": "A bounded monotone tail and the supremum argument"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-monotone-sequence-convergence-theorem-01",
    "source": {
      "mko_id": "mko-monotone-sequence-convergence-theorem",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "7347d54ee6a5bad0a10d31d2740a6f5af3d2539f9dd91cfc6e85d51ea7c966dc",
      "english_sha256": "04ac52a1866dfc205ff11c701b766a16b5e2bfa894396f50a16b4f6eb495d448"
    },
    "mode": "theorem_proof",
    "audience_prerequisites": [
      "algebra",
      "epsilon",
      "supremum",
      "definition_here"
    ],
    "goal": {
      "zh-Hant": "對 a_n=3-2/(n+1) 證單調、有界、sup=3 及極限。",
      "en": "For a_n=3-2/(n+1), prove monotonicity, boundedness, supremum 3 and convergence."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "a_n=3-2/(n+1) 對全部 n≥0 定義；使用實數上確界性。",
          "en": "a_n=3-2/(n+1) for every n≥0; use the least-upper-bound property of R."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "a_{n+1}-a_n=2/(n+1)-2/(n+2)",
        "after_tex": "a_{n+1}-a_n=2/((n+1)(n+2))>0",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [
          "n\\ge0"
        ],
        "explanation": {
          "zh-Hant": "兩分母正，證每個相鄰差正；因此全序列嚴格遞增。",
          "en": "Both denominators are positive, proving every adjacent difference positive and hence whole-sequence strict increase."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "對所有 n 的公式，不是有限相鄰抽查。",
            "en": "The formula holds for every n, not sampled neighbors."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S02",
        "uses": [
          "G1"
        ],
        "before_tex": "0<2/(n+1)\\le2",
        "after_tex": "1\\le a_n<3",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "給下界 1 與適用遞增定理的上界 3。",
          "en": "This gives lower bound 1 and the upper bound 3 needed for the increasing-sequence theorem."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "上界不是最大值，因每項嚴格小於 3。",
            "en": "The upper bound is not a maximum because every term is strictly below 3."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02"
        ],
        "before_tex": "\\varepsilon>0,\\quad N=\\lfloor2/\\varepsilon\\rfloor+1",
        "after_tex": "0<3-a_n=2/(n+1)<\\varepsilon\\quad(n\\ge N)",
        "relation": "implies",
        "rule": "epsilon",
        "conditions": [],
        "explanation": {
          "zh-Hant": "全部尾端分母更大，直接證 a_n→3。",
          "en": "Every later denominator is larger, directly proving a_n tends to 3."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "N 為一個固定尾端索引，之後對所有 n 使用。",
            "en": "N is fixed once for the entire later tail."
          }
        },
        "leaf_basis": "epsilon"
      },
      {
        "id": "S04",
        "uses": [
          "S02",
          "S03"
        ],
        "before_tex": "u<3,\\quad\\varepsilon=3-u>0",
        "after_tex": "a_N>3-\\varepsilon=u;\\quad\\sup\\{a_n:n\\ge0\\}=3",
        "relation": "implies",
        "rule": "supremum",
        "conditions": [],
        "explanation": {
          "zh-Hant": "任何小於 3 的數都不是上界，結合上界資格得到最小性。",
          "en": "Every number below 3 fails as an upper bound; combined with the upper-bound check this gives leastness."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "sup 不需是某項，前面已證無項等於 3。",
            "en": "A supremum need not be a term; no term equals 3 here."
          }
        },
        "leaf_basis": "supremum"
      },
      {
        "id": "S05",
        "uses": [
          "G1"
        ],
        "before_tex": "(u_n)\\text{ nondecreasing and bounded above},\\quad s=\\sup\\{u_n\\}",
        "after_tex": "\\forall\\varepsilon>0\\ \\exists N:\\ s-\\varepsilon<u_N\\le s",
        "relation": "implies",
        "rule": "supremum",
        "conditions": [
          "(u_n)\\text{ nondecreasing and bounded above}"
        ],
        "explanation": {
          "zh-Hant": "另示一般定理核心：s-epsilon 非上界，故有某項超過。",
          "en": "For the general theorem's core, s-epsilon is not an upper bound, so some term exceeds it."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "這裡明示另取一般 u_n 及其兩項前提，不與具體 a_n 混名。",
            "en": "This separately names a general u_n and both premises, avoiding symbol collision with a_n."
          }
        },
        "leaf_basis": "supremum"
      },
      {
        "id": "S06",
        "uses": [
          "S05"
        ],
        "before_tex": "n\\ge N",
        "after_tex": "s-\\varepsilon<u_N\\le u_n\\le s",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "全稱單調性把一項的接近轉成全部尾端的接近。",
          "en": "Whole-sequence monotonicity transfers one close term to the entire tail."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S05"
          ],
          "explanation": {
            "zh-Hant": "缺單調性就不能從 u_N 推全部後續項。",
            "en": "Without monotonicity, u_N cannot control all later terms."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S07",
        "uses": [
          "S06"
        ],
        "before_tex": "0\\le s-u_n<\\varepsilon\\quad(n\\ge N)",
        "after_tex": "u_n\\to s",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "絕對誤差就是 s-u_n，符合極限定義，完成一般上確界論證。",
          "en": "The absolute error equals s-u_n, meeting the limit definition and completing the supremum proof."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S06"
          ],
          "explanation": {
            "zh-Hant": "只用所列單調與有界前提，不從有限樣本推前提。",
            "en": "Only the stated monotonicity and bound premises are used, not inferred from samples."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S08",
        "uses": [
          "S01",
          "S02",
          "S04",
          "S07"
        ],
        "before_tex": "a_n\\text{ nondecreasing},\\quad\\sup\\{a_n\\}=3",
        "after_tex": "\\lim a_n=3\\text{, not attained by any }a_n",
        "relation": "implies",
        "rule": "epsilon",
        "conditions": [],
        "explanation": {
          "zh-Hant": "一般證法與具體尾端給同一答案，並保留未取到邊界。",
          "en": "The general proof and direct tail agree on the limit, retaining the nonattainment boundary."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01",
            "S02",
            "S04",
            "S07"
          ],
          "explanation": {
            "zh-Hant": "沒有反推所有收斂數列都單調。",
            "en": "No converse claiming all convergent sequences monotone is inferred."
          }
        },
        "leaf_basis": "epsilon"
      }
    ],
    "result": {
      "zh-Hant": "a_n 嚴格遞增有界，趨 3；3 是未取到的 sup。",
      "en": "a_n is strictly increasing and bounded, tends to 3, and has unattained supremum 3."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "a_n 有最大值 3。",
          "en": "a_n has maximum 3."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "每項均嚴格小於 3。",
          "en": "Every term is strictly below 3."
        }
      },
      {
        "claim": {
          "zh-Hant": "單調就一定收斂成實數。",
          "en": "Monotonicity alone gives a real limit."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "n 遞增但無上界，無有限極限。",
          "en": "n increases without an upper bound and has no finite limit."
        }
      },
      {
        "claim": {
          "zh-Hant": "有界就一定收斂。",
          "en": "Boundedness alone gives convergence."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "(-1)^n 有界而偶奇極限不同。",
          "en": "(-1)^n is bounded with different parity limits."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "保留全稱monotonicity與相應bound。若s=sup{a_n}，s-epsilon不是upper bound給出N，之後由monotonicity控制全部tail。sup/inf未必是某一項。有限prefix不證全序列，monotone無界或bounded nonmonotone不在定理假設。",
          "en": "Keep whole-sequence monotonicity and the corresponding bound. If s is the supremum of the terms, s−epsilon is not an upper bound, supplying an index N; monotonicity then controls every later term. A supremum or infimum need not be a term. A finite prefix does not supply a universal certificate. A monotone sequence without the required bound, or a bounded sequence lacking monotonicity, does not meet this theorem's hypotheses."
        },
        "source": {
          "mko_id": "mko-monotone-sequence-convergence-theorem",
          "sha256": "7347d54ee6a5bad0a10d31d2740a6f5af3d2539f9dd91cfc6e85d51ea7c966dc",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "algebra",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "實數域的等式運算；除法前分母必非零，不等式乘正數保持方向。",
          "en": "Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities."
        }
      },
      {
        "id": "epsilon",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "收斂表示每個正 epsilon 有一個整數 N，使全部 n≥N 的誤差嚴格小於 epsilon；使用實數 Archimedean 性。",
          "en": "Convergence requires for every positive epsilon an integer N controlling every n≥N by a strict error bound; use the Archimedean property of the reals."
        }
      },
      {
        "id": "supremum",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "非空有上界實集合有上確界 s；若 epsilon>0，s-epsilon 不是上界。",
          "en": "A nonempty bounded-above real set has supremum s; for epsilon>0, s-epsilon is not an upper bound."
        }
      },
      {
        "id": "definition_here",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "實數數列極限 L 的 epsilon 定義：每個正 epsilon 都有 N，使所有 n≥N 的 |u_n-L|<epsilon。",
          "en": "The epsilon definition of real-sequence limit L requires, for each positive epsilon, an N with |u_n-L|<epsilon for every n≥N."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "monotone-difference",
        "lhs": "2/6-2/7",
        "rhs": "2/(6*7)",
        "step": "S01"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "完整 n≥0 公式、嚴格 epsilon 證書；一般定理的 supremum 步驟另明示。",
      "en": "A full n≥0 law with a strict epsilon certificate; explicitly include the general theorem's supremum step."
    },
    "md_symbols": [
      [
        "a_n",
        "具體全數列",
        "concrete whole sequence"
      ],
      [
        "s",
        "一般定理上確界",
        "general theorem supremum"
      ]
    ],
    "md_error_controls": [
      {
        "title": "前提成對 / Paired premises",
        "zh": "遞增配上界、遞減配下界。",
        "en": "Pair nondecreasing with an upper bound and nonincreasing with a lower bound."
      },
      {
        "title": "嚴格界 / Strict bound",
        "zh": "s-epsilon 非上界提供嚴格的近似項。",
        "en": "Failure of s-epsilon as an upper bound supplies a strictly close term."
      }
    ],
    "md_open_items": []
  },
  "revision": {
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    "collection": "coverage-20261021",
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    "original_source_sha256": "318f43093b381e02533751c73966ee4bcfc677e16c35ebd8ca54ffcfe2049934",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [
      "mko_definition"
    ],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
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    ],
    "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a",
    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite.",
    "english_binding_revision": {
      "id": "edition-symbol-bindings-20261101",
      "path": "content/edition-symbol-bindings-20261101/revision.json",
      "sha256": "e9b4c69f2c4732074c37abb624c8ab8a70afc3645c12bf9df2268ef650ad7b6c"
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  }
}