{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-monotone-sequence-convergence-theorem",
  "semantic_family_id": "sf-monotone-sequence-convergence-theorem",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "單調有界數列收斂定理"
  },
  "summary": {
    "zh-Hant": "非遞減且有上界的實數列收斂至其項集合supremum；非遞增版本對偶收斂至infimum。"
  },
  "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。"
  },
  "formula": {
    "tex": "a_n\\to s",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>→</mo><mi>s</mi></mrow></math>",
    "semantic_ast": {
      "type": "tends_to",
      "expression": {
        "type": "subscript",
        "base": {
          "type": "symbol",
          "name": "a",
          "glyph": "a"
        },
        "subscript": {
          "type": "symbol",
          "name": "n",
          "glyph": "n"
        }
      },
      "target": {
        "type": "symbol",
        "name": "s",
        "glyph": "s"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "3fb3b2e1874c8d4ad586fd85d41ff09b7119f79e32d074c6b9afb26ad28103d6"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "主證書使用a_n=L±c/(n+s)，c>0且s>=1的symbolic全tail。"
    },
    {
      "id": "a2",
      "zh-Hant": "epsilon為正有理數並求嚴格誤差witness。"
    }
  ],
  "symbols": [
    {
      "token": "a_n",
      "role": "monotone_real_sequence",
      "role_zh": "單調實數列",
      "scope": "representative_formula"
    },
    {
      "token": "s",
      "role": "supremum_or_infimum",
      "role_zh": "極限界",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-supremum-and-infimum",
      "title_zh": "上確界與下確界",
      "reason_zh": "極限候選。"
    },
    {
      "id": "mko-sequence-limit",
      "title_zh": "數列極限",
      "reason_zh": "epsilon-tail。"
    },
    {
      "id": "mko-order-relation",
      "title_zh": "偏序與全序",
      "reason_zh": "monotonicity與bounds。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "nondecreasing配upper bound，nonincreasing配lower bound。",
      "sup/inf未必attained。",
      "finite prefix與缺一hypothesis皆不足。"
    ],
    "common_misconception_zh": "由幾項單調便宣稱全序列收斂，或把sup一律當maximum。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "exact families限明示symbolic tails；不由有限samples證一般定理。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入Bolzano-Weierstrass與更多compactness結果。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-monotone-sequence-convergence-theorem.md 整理；2026-10-14 脈衝 R02（使用者手動接續 2026-11-08 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}