{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "4SNJWQBAHF"
    },
    {
      "field": "about",
      "target": "P8TMYNPSYD"
    },
    {
      "field": "uses",
      "target": "P8TMYNPSYD"
    }
  ],
  "body": "\n## Summary\n\nA bounded real sequence is almost convergent to $L$ when every [[Banach limit|SYF2ND6J97]] gives it the value $L$.\nEvery convergent sequence is almost convergent to its limit, but some divergent sequences, such as $1, 0, 1, 0, \\dots$, are almost convergent too.\n[[Lorentz's characterization|P8TMYNPSYD]] describes almost convergence by averages over long blocks of consecutive terms.\n\n## Explanations\n\n### L1 human\n\nBanach limits are ways to assign a \"limit\" to every bounded sequence, and there are many of them.\nFor most divergent sequences they disagree with each other.\nA sequence is almost convergent when all of them agree on one value.\nFor $1, 0, 1, 0, \\dots$ they all give $1/2$, which is the average value the sequence takes in the long run, uniformly along the sequence.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "40H05",
      "46A45"
    ],
    "name": "Almost convergence",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Banach_limit"
      }
    ],
    "tag": "8HTHB1QPZY"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/8HTHB1QPZY/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/8HTHB1QPZY.md",
  "msc": [
    "40H05",
    "46A45"
  ],
  "name": "Almost convergence",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/8HT/8HTHB1QPZY.md",
  "status": null,
  "tag": "8HTHB1QPZY",
  "uri": "m2://8HTHB1QPZY",
  "url": "https://math2.tamnd.com/wiki/almost-convergence/",
  "writing": {
    "tag": "8HTHB1QPZY",
    "kind": "concept",
    "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
      }
    ]
  }
}