{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $(a_n)$ be a bounded real sequence.\nThen $(a_n)$ converges if and only if $\\liminf_{n \\to \\infty} a_n = \\limsup_{n \\to \\infty} a_n$.\nIn this case, $\\lim_{n \\to \\infty} a_n$ is equal to this common value.\n\n## Remarks\n\nThe proof of the direction from right to left uses $\\inf_{k \\ge n} a_k \\le a_n \\le \\sup_{k \\ge n} a_k$ and the [[squeeze theorem|0XMF00T4XQ]].\nIn the extended real line, the statement holds for every sequence. Then convergence to $+\\infty$ or to $-\\infty$ is allowed.\nFor $(-1)^n$, the two values are $-1$ and $1$. They differ, so the sequence diverges.\n\n## Explanations\n\n### L1 human\n\nThe limit inferior and the limit superior are the lowest and the highest points where the sequence keeps returning.\nIf these two points are the same, the sequence has nowhere else to go, so it converges to that point.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "40A05"
    ],
    "name": "A sequence converges iff its liminf and limsup agree",
    "role": "proposition",
    "sources": [
      {
        "locator": "Properties",
        "url": "https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior"
      }
    ],
    "tag": "0HM732ZNVC",
    "uses": [
      "0H4R3RESGV",
      "0P8BA7T7S5",
      "0AZ3R52593",
      "0XMF00T4XQ"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0HM732ZNVC/",
  "links": [
    {
      "field": "uses",
      "target": "0H4R3RESGV"
    },
    {
      "field": "uses",
      "target": "0P8BA7T7S5"
    },
    {
      "field": "uses",
      "target": "0AZ3R52593"
    },
    {
      "field": "uses",
      "target": "0XMF00T4XQ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0HM732ZNVC.md",
  "msc": [
    "40A05"
  ],
  "name": "A sequence converges iff its liminf and limsup agree",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0HM/0HM732ZNVC.md",
  "status": null,
  "tag": "0HM732ZNVC",
  "uri": "m2://0HM732ZNVC",
  "url": "https://math2.tamnd.com/wiki/a-sequence-converges-iff-its-liminf-and-limsup-agree/",
  "writing": {
    "tag": "0HM732ZNVC",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}