{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\n1. Let $\\varepsilon \u003e 0$.\n2. Since $f$ is [[uniformly continuous|0Q4DR37DVE]], there is $\\delta \u003e 0$ with $\\abs{f(x) - f(y)} \u003c \\varepsilon$ for all $x, y \\in D$ with $\\abs{x - y} \u003c \\delta$.\n3. Since $(x_n)$ is a [[Cauchy sequence|0QZMTN5AXX]], there is $N$ with $\\abs{x_m - x_n} \u003c \\delta$ for all $m, n \\ge N$.\n4. Let $m, n \\ge N$. The points $x_m$ and $x_n$ lie in $D$ and satisfy $\\abs{x_m - x_n} \u003c \\delta$. So step 2 gives $\\abs{f(x_m) - f(x_n)} \u003c \\varepsilon$.\n5. Since $\\varepsilon \u003e 0$ was arbitrary, $(f(x_n))$ is a Cauchy sequence.\n\n## Idea\n\nUniform continuity turns \"inputs within $\\delta$\" into \"outputs within $\\varepsilon$\", with one $\\delta$ for all points. A Cauchy sequence eventually has all its terms within $\\delta$ of each other. So the images are eventually within $\\varepsilon$ of each other.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "26A15"
    ],
    "name": "Proof that uniformly continuous functions map Cauchy sequences to Cauchy sequences",
    "proves": "026FDZFRXS",
    "sources": [
      {
        "locator": "Lemma 3.4.5",
        "url": "https://www.jirka.org/ra/html/sec_unifcont.html"
      }
    ],
    "tag": "19VV6JCYN4",
    "uses": [
      "0Q4DR37DVE",
      "0QZMTN5AXX"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/19VV6JCYN4/",
  "links": [
    {
      "field": "proves",
      "target": "026FDZFRXS"
    },
    {
      "field": "uses",
      "target": "0Q4DR37DVE"
    },
    {
      "field": "uses",
      "target": "0QZMTN5AXX"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/19VV6JCYN4.md",
  "msc": [
    "26A15"
  ],
  "name": "Proof that uniformly continuous functions map Cauchy sequences to Cauchy sequences",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/19V/19VV6JCYN4.md",
  "status": null,
  "tag": "19VV6JCYN4",
  "uri": "m2://19VV6JCYN4",
  "url": "https://math2.tamnd.com/wiki/proof-that-uniformly-continuous-functions-map-cauchy-sequences-to-cauchy-sequences/",
  "writing": {
    "tag": "19VV6JCYN4",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}