{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $(a_n)$ be a real sequence that converges to a real number $L$.\nLet $(a_{n_k})_{k \\in \\mathbb{N}}$ be a [[subsequence|0R8J3ZQMJS]] of $(a_n)$.\nThen $a_{n_k} \\to L$ as $k \\to \\infty$.\n\n## Remarks\n\nFor the proof, let $\\varepsilon \u003e 0$ and take $N$ with $|a_n - L| \u003c \\varepsilon$ for $n \\ge N$. Since $n_k \\ge k$, we get $|a_{n_k} - L| \u003c \\varepsilon$ for all $k \\ge N$.\nIt gives a test for divergence. If two subsequences have different limits, the sequence diverges. For $(-1)^n$ the even terms tend to $1$ and the odd terms tend to $-1$.\n\n## Explanations\n\n### L1 human\n\nIf all terms eventually stay close to $L$, then the terms that you keep also stay close to $L$.\nDeleting terms cannot change where the sequence is going.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "40A05"
    ],
    "name": "Subsequences of a convergent sequence converge to the same limit",
    "role": "proposition",
    "sources": [
      {
        "locator": "Properties",
        "url": "https://en.wikipedia.org/wiki/Limit_of_a_sequence"
      },
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Subsequence"
      }
    ],
    "tag": "0VHAVXW08V",
    "uses": [
      "0R8J3ZQMJS",
      "0XKWSSY8P2",
      "09G6KHS6GY"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0VHAVXW08V/",
  "links": [
    {
      "field": "uses",
      "target": "0R8J3ZQMJS"
    },
    {
      "field": "uses",
      "target": "0XKWSSY8P2"
    },
    {
      "field": "uses",
      "target": "09G6KHS6GY"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0VHAVXW08V.md",
  "msc": [
    "40A05"
  ],
  "name": "Subsequences of a convergent sequence converge to the same limit",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0VH/0VHAVXW08V.md",
  "status": null,
  "tag": "0VHAVXW08V",
  "uri": "m2://0VHAVXW08V",
  "url": "https://math2.tamnd.com/wiki/subsequences-of-a-convergent-sequence-converge-to-the-same-limit/",
  "writing": {
    "tag": "0VHAVXW08V",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}