{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Example\n\nLet $a_n = 1/n$ for each integer $n \\ge 1$.\nThen $a_n \\to 0$ as $n \\to \\infty$.\nLet $\\varepsilon \u003e 0$. By the Archimedean property of $\\mathbb{R}$, there is a natural number $N$ with $N \u003e 1/\\varepsilon$.\nFor each $n \\ge N$ we get $|a_n - 0| = 1/n \\le 1/N \u003c \\varepsilon$.\nBy the [[epsilon-N definition|09G6KHS6GY]], this shows $1/n \\to 0$.\n\n## Remarks\n\nThe proof uses the Archimedean property, which says that the natural numbers are not bounded above in $\\mathbb{R}$.\nWith the [[algebra of limits|07XBM3A9J5]], it gives many more limits, for example $1/n^2 \\to 0$ and $(n + 1)/n \\to 1$.\nMathlib states it for $1/n$ over $\\mathbb{R}$ as `tendsto_one_div_atTop_nhds_zero_nat`.\n\n## Explanations\n\n### L1 human\n\nThe numbers $1, 1/2, 1/3, 1/4, \\dots$ get smaller and smaller, and they approach $0$.\nTo get $1/n$ below $0.001$, it is enough to take $n \u003e 1000$.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "instance_of": [
      "0W4CN9E2W5"
    ],
    "kind": "example",
    "msc": [
      "40A05"
    ],
    "name": "The sequence 1/n tends to zero",
    "sources": [
      {
        "locator": "Examples",
        "url": "https://en.wikipedia.org/wiki/Limit_of_a_sequence"
      }
    ],
    "tag": "08QAX4H1PF",
    "uses": [
      "0XKWSSY8P2",
      "09G6KHS6GY"
    ]
  },
  "kind": "example",
  "link": "https://math2.tamnd.com/node/08QAX4H1PF/",
  "links": [
    {
      "field": "instance_of",
      "target": "0W4CN9E2W5"
    },
    {
      "field": "uses",
      "target": "0XKWSSY8P2"
    },
    {
      "field": "uses",
      "target": "09G6KHS6GY"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/08QAX4H1PF.md",
  "msc": [
    "40A05"
  ],
  "name": "The sequence 1/n tends to zero",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/08Q/08QAX4H1PF.md",
  "status": null,
  "tag": "08QAX4H1PF",
  "uri": "m2://08QAX4H1PF",
  "url": "https://math2.tamnd.com/wiki/the-sequence-1-n-tends-to-zero/",
  "writing": {
    "tag": "08QAX4H1PF",
    "kind": "example",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}