{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-bolzano-weierstrass-theorem-real-01",
  "mko_id": "mko-bolzano-weierstrass-theorem-real",
  "title": {
    "zh-Hant": "振盪尾的兩個收斂子數列",
    "en": "Two convergent subsequences of an oscillatory tail"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-bolzano-weierstrass-theorem-real-01",
    "source": {
      "mko_id": "mko-bolzano-weierstrass-theorem-real",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "68509f22469c3dc3548c8128faec469688cb7a72f7b9fca9c4539568db8e44ea",
      "english_sha256": "484aa7b44b137b24493d954852772e36bbac420461a485eeb02b763116637432"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "algebra",
      "epsilon"
    ],
    "goal": {
      "zh-Hant": "對 a_n=(-1)^n+1/(n+1) 給有界性與兩個完整收斂子數列。",
      "en": "For a_n=(-1)^n+1/(n+1), provide boundedness and two complete convergent subsequences."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "a_n=(-1)^n+1/(n+1)，n≥0；索引取 k(j)=2j、ell(j)=2j+1。",
          "en": "a_n=(-1)^n+1/(n+1), n≥0; take k(j)=2j and ell(j)=2j+1."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "|(-1)^n|=1,\\quad0<1/(n+1)\\le1",
        "after_tex": "|a_n|\\le2\\quad\\forall n\\ge0",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "三角界給同一 M=2 控制全部項。",
          "en": "The triangle bound gives one M=2 controlling every term."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "不是以 prefix 最大值代替全稱有界性。",
            "en": "A prefix maximum is not substituted for whole-sequence boundedness."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S02",
        "uses": [
          "G1"
        ],
        "before_tex": "k(j+1)-k(j)=2,\\quad\\ell(j+1)-\\ell(j)=2",
        "after_tex": "k,\\ell:\\mathbb N_0\\to\\mathbb N_0\\text{ strictly increasing}",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩個仿射完整索引均步長正且起點非負。",
          "en": "Both complete affine indices have positive step and nonnegative start."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "無限定義域已明示，不是有限索引列表。",
            "en": "The infinite index domain is specified, not a finite list."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02"
        ],
        "before_tex": "a_{k(j)}=a_{2j}",
        "after_tex": "a_{2j}=1+1/(2j+1)",
        "relation": "equivalent",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "偶指數符號為正，先代索引再算值。",
          "en": "Even parity gives a positive sign; substitute the index before evaluating."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "G1",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "分母是 2j+1，不是 2j。",
            "en": "The denominator is 2j+1, not 2j."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S04",
        "uses": [
          "S03"
        ],
        "before_tex": "\\varepsilon>0,\\quad J=\\lfloor1/\\varepsilon\\rfloor+1",
        "after_tex": "j\\ge J\\Longrightarrow |a_{2j}-1|=1/(2j+1)<\\varepsilon",
        "relation": "implies",
        "rule": "epsilon",
        "conditions": [],
        "explanation": {
          "zh-Hant": "共同分母下界給全部偶子數列尾端趨 1。",
          "en": "The denominator lower bound proves the whole even-subsequence tail tends to 1."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03"
          ],
          "explanation": {
            "zh-Hant": "嚴格界對所有 j≥J。",
            "en": "The strict bound holds for every j≥J."
          }
        },
        "leaf_basis": "epsilon"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S02"
        ],
        "before_tex": "a_{\\ell(j)}=a_{2j+1}",
        "after_tex": "a_{2j+1}=-1+1/(2j+2)",
        "relation": "equivalent",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "奇指數符號為負，仍保留正的有理擾動。",
          "en": "Odd parity gives a negative sign while retaining the positive rational perturbation."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "G1",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "極限為 -1 而不是絕對值 1。",
            "en": "The limit is -1, not its magnitude 1."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S06",
        "uses": [
          "S05"
        ],
        "before_tex": "j\\ge J=\\lfloor1/\\varepsilon\\rfloor+1",
        "after_tex": "|a_{2j+1}+1|=1/(2j+2)<\\varepsilon",
        "relation": "implies",
        "rule": "epsilon",
        "conditions": [],
        "explanation": {
          "zh-Hant": "同一 J 亦足夠控制奇子數列的所有尾項。",
          "en": "The same J also controls every term in the odd-subsequence tail."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S05"
          ],
          "explanation": {
            "zh-Hant": "這是獨立核奇序列，非從偶序列推相同極限。",
            "en": "This independently checks the odd subsequence, rather than inferring the even limit for it."
          }
        },
        "leaf_basis": "epsilon"
      },
      {
        "id": "S07",
        "uses": [
          "S01",
          "S02",
          "S04",
          "S06"
        ],
        "before_tex": "a_{2j}\\to1,\\quad a_{2j+1}\\to-1",
        "after_tex": "(a_n)\\text{ has convergent subsequences with different limits}",
        "relation": "implies",
        "rule": "epsilon",
        "conditions": [],
        "explanation": {
          "zh-Hant": "符合 BW 存在某個子數列的結論，此具體模型還給出兩個。",
          "en": "This supplies the existence promised by BW in this instance, and in fact gives two subsequences."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01",
            "S02",
            "S04",
            "S06"
          ],
          "explanation": {
            "zh-Hant": "實例論證沒有聲稱所有有界數列都已由有限測試證明。",
            "en": "The instance does not claim the theorem for all bounded sequences from finite tests."
          }
        },
        "leaf_basis": "epsilon"
      },
      {
        "id": "S08",
        "uses": [
          "S04",
          "S06"
        ],
        "before_tex": "1\\ne-1",
        "after_tex": "(a_n)\\text{ does not converge}",
        "relation": "implies",
        "rule": "epsilon",
        "conditions": [],
        "explanation": {
          "zh-Hant": "若原數列收斂至 L，任意嚴格遞增索引最終超過原 N，兩子數列都應趨 L；極限唯一性矛盾。",
          "en": "If the original converged to L, every strictly increasing index would eventually exceed its tail N, forcing both subsequences to L; uniqueness of limits contradicts this."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S04",
            "S06"
          ],
          "explanation": {
            "zh-Hant": "只由不同已證子極限否定原收斂，不因振盪外觀猜測。",
            "en": "Nonconvergence follows from different proved subsequential limits, not visual oscillation."
          }
        },
        "leaf_basis": "epsilon"
      }
    ],
    "result": {
      "zh-Hant": "全序列有界，偶子列趨 1、奇子列趨 -1，所以原列不收斂。",
      "en": "The whole sequence is bounded; the even subsequence tends to 1 and the odd to -1, so the original does not converge."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "BW 保證原數列收斂。",
          "en": "BW guarantees original-sequence convergence."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "本例兩個已證子極限不同。",
          "en": "This example has two different proved subsequential limits."
        }
      },
      {
        "claim": {
          "zh-Hant": "所有收斂子數列極限必相同。",
          "en": "All convergent subsequences must have the same limit."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "此例的 1 與 -1 不同。",
          "en": "The limits 1 and -1 here differ."
        }
      },
      {
        "claim": {
          "zh-Hant": "看到有限重複值就證 BW。",
          "en": "Finite repeated values prove BW."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "仍需完整無限索引、全序列界及全尾證書。",
          "en": "Complete infinite indices, a whole-sequence bound and an all-tail certificate are required."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "bounded 須為 exists M forall n；subsequence index genuinely strictly increasing 且無限，並有 forall-tail epsilon witness。週期列與 oscillatory rational tails 提供 exact slices。有限 prefix／cluster search 不足；定理不保證原數列收斂，也不保證不同收斂子數列的極限相同。",
          "en": "Boundedness means there exists M bounding all terms, not just a finite prefix. A subsequence has an infinite strictly increasing index map, and convergence has an all-later epsilon-tail witness. Periodic sequences and symbolic oscillatory rational tails give exact instances here. A finite prefix or finite cluster search is insufficient. The theorem does not require the original sequence to converge or guarantee that every convergent subsequence has the same limit."
        },
        "source": {
          "mko_id": "mko-bolzano-weierstrass-theorem-real",
          "sha256": "68509f22469c3dc3548c8128faec469688cb7a72f7b9fca9c4539568db8e44ea",
          "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."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "even-tail",
        "lhs": "1/23",
        "rhs": "1/10",
        "step": "S04",
        "relation": "<"
      },
      {
        "id": "odd-tail",
        "lhs": "1/24",
        "rhs": "1/10",
        "step": "S06",
        "relation": "<"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "只證宣告的符號數列實例，不以其替代一般 BW 定理。",
      "en": "Only the declared symbolic sequence, not a replacement for the general BW theorem."
    },
    "md_symbols": [
      [
        "k,\\ell",
        "偶奇索引",
        "parity indices"
      ],
      [
        "M",
        "全序列界",
        "whole-sequence bound"
      ]
    ],
    "md_error_controls": [
      {
        "title": "全序列有界 / Whole-sequence bound",
        "zh": "M 必對每個 n 成立。",
        "en": "M must hold for every n."
      },
      {
        "title": "子列三責任 / Three subsequence obligations",
        "zh": "索引嚴格遞增、無限、且有全尾收斂界。",
        "en": "Indices must be strictly increasing and infinite, with an all-tail convergence bound."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
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    "original_source_sha256": "605ab67bfeb763ae5cddd56fa58c8fa84b2fbce158ebd93195090c891bdf3c42",
    "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."
  }
}