{
  "aliases": [
    "limsup"
  ],
  "backlinks": [
    {
      "field": "uses",
      "target": "05H4DDDYBZ"
    },
    {
      "field": "uses",
      "target": "07E5MB1CA2"
    },
    {
      "field": "uses",
      "target": "0HM732ZNVC"
    }
  ],
  "body": "\n## Definition\n\nLet $(a_n)$ be a [[bounded sequence|05FQ3VGAR9]] of real numbers.\nFor each $n \\in \\mathbb{N}$, let $s_n = \\sup \\{a_k : k \\ge n\\}$.\nThe sequence $(s_n)$ is decreasing and bounded below.\nThe limit superior of $(a_n)$ is\n$$\\limsup_{n \\to \\infty} a_n = \\lim_{n \\to \\infty} s_n = \\inf_{n \\in \\mathbb{N}} \\sup_{k \\ge n} a_k.$$\n\n## Remarks\n\nThe limit exists because a [[decreasing sequence bounded below converges|02V2GYAC8P]].\nFor a sequence that is not bounded above, one sets $\\limsup a_n = +\\infty$. For a sequence that is bounded above and not below, the same formula is used in the extended real line, and it can give $-\\infty$.\nFor $a_n = (-1)^n (1 + 1/(n+1))$, the limit superior is $1$, although no term equals $1$.\nFor a bounded sequence, it is the largest [[cluster point|0RCKQN1E09]].\n\n## Explanations\n\n### L1 human\n\nThe limit superior is the highest value that the sequence keeps coming back to, in the long run.\nEach number above it is exceeded by only finitely many terms.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "aliases": [
      "limsup"
    ],
    "defines": "0T3RZQXDYD",
    "kind": "definition",
    "msc": [
      "40A05"
    ],
    "name": "Limit superior",
    "sources": [
      {
        "locator": "The case of sequences of real numbers",
        "url": "https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior"
      }
    ],
    "tag": "0P8BA7T7S5",
    "uses": [
      "05FQ3VGAR9"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0P8BA7T7S5/",
  "links": [
    {
      "field": "defines",
      "target": "0T3RZQXDYD"
    },
    {
      "field": "uses",
      "target": "05FQ3VGAR9"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0P8BA7T7S5.md",
  "msc": [
    "40A05"
  ],
  "name": "Limit superior",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0P8/0P8BA7T7S5.md",
  "status": null,
  "tag": "0P8BA7T7S5",
  "uri": "m2://0P8BA7T7S5",
  "url": "https://math2.tamnd.com/wiki/limit-superior-definition/",
  "writing": {
    "tag": "0P8BA7T7S5",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}