{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "CWP26Y1TCZ"
    },
    {
      "field": "uses",
      "target": "MZM1AFXY82"
    },
    {
      "field": "uses",
      "target": "WW53KG71KH"
    }
  ],
  "body": "\n## Definition\n\nLet $X$ be a set.\nA double sequence in $X$ is a [[function|0EPAT51DXQ]] $x \\colon \\N \\times \\N \\to X$.\nIts value at $(m, n)$ is written $x_{m,n}$, and the double sequence is written $(x_{m,n})_{m, n \\in \\N}$.\n\n## Remarks\n\nA double sequence is a [[sequence|0QE8STGJH2]] indexed by $\\N \\times \\N$ instead of $\\N$.\nExample: $x_{m,n} = m/(m + n + 1)$ is a double sequence of real numbers.\nFor fixed $m$ it tends to $0$ as $n \\to \\infty$, and for fixed $n$ it tends to $1$ as $m \\to \\infty$.\nSo $\\lim_m \\lim_n x_{m,n} = 0$ and $\\lim_n \\lim_m x_{m,n} = 1$: the two iterated limits differ.\nNon-example: a triangular array $(x_{m,n})_{0 \\le n \\le m}$ is not a double sequence, since it is defined only for $n \\le m$; it becomes one after extension by $0$.\nThe joint limit in the sense of [[Pringsheim convergence|QYTGCW1GTK]] is the usual notion of limit for double sequences.\n\n## Explanations\n\n### L1 human\n\nA double sequence assigns an element to each pair $(m, n)$ of natural numbers, like an infinite matrix.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "B70G8FYPG0",
    "kind": "definition",
    "msc": [
      "40B05"
    ],
    "name": "Double sequence",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Iterated_limit"
      }
    ],
    "tag": "WK5V372TG3",
    "uses": [
      "0QE8STGJH2",
      "0EPAT51DXQ"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/WK5V372TG3/",
  "links": [
    {
      "field": "defines",
      "target": "B70G8FYPG0"
    },
    {
      "field": "uses",
      "target": "0QE8STGJH2"
    },
    {
      "field": "uses",
      "target": "0EPAT51DXQ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/WK5V372TG3.md",
  "msc": [
    "40B05"
  ],
  "name": "Double sequence",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/WK5/WK5V372TG3.md",
  "status": null,
  "tag": "WK5V372TG3",
  "uri": "m2://WK5V372TG3",
  "url": "https://math2.tamnd.com/wiki/double-sequence-definition/",
  "writing": {
    "tag": "WK5V372TG3",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}