{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\nWe prove the forward direction directly and the converse by contraposition.\n\n1. Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]] and let $f \\colon X \\to Y$.\n2. By definition, $f$ is [[continuous|0SY9ECQ19Q]] when $f^{-1}(V)$ is open in $X$ for each open $V$ of $Y$.\n3. By [[continuity via open sets|0YF82GBTND]], $f$ is continuous if and only if condition (C) holds. (C): for each $a \\in X$ and $\\varepsilon \u003e 0$ there is $\\delta \u003e 0$ such that $d_X(x, a) \u003c \\delta$ implies $d_Y(f(x), f(a)) \u003c \\varepsilon$.\n4. Suppose $f$ is continuous. Let $(x_n)$ be a sequence that [[converges|06JYPJ4WHH]] to $x \\in X$, and let $\\varepsilon \u003e 0$.\n5. By (C) at $a = x$, there is $\\delta \u003e 0$ such that $d_X(z, x) \u003c \\delta$ implies $d_Y(f(z), f(x)) \u003c \\varepsilon$.\n6. Since $x_n \\to x$, there is an integer $N$ with $d_X(x_n, x) \u003c \\delta$ for all $n \\ge N$.\n7. By step 5, $d_Y(f(x_n), f(x)) \u003c \\varepsilon$ for all $n \\ge N$. Hence $f(x_n) \\to f(x)$.\n8. Conversely, suppose $f$ is not continuous. By step 3, (C) fails at some $a \\in X$.\n9. So there is $\\varepsilon \u003e 0$ such that each $\\delta \u003e 0$ has some $x \\in X$ with $d_X(x, a) \u003c \\delta$ and $d_Y(f(x), f(a)) \\ge \\varepsilon$.\n10. For each integer $n \\ge 1$, apply step 9 with $\\delta = 1/n$. This gives $x_n \\in X$ with $d_X(x_n, a) \u003c 1/n$ and $d_Y(f(x_n), f(a)) \\ge \\varepsilon$.\n11. We show $x_n \\to a$. Let $\\eta \u003e 0$. The [[reals are Archimedean|0A3GWZ86AY]], so some natural number $N$ has $N \\eta \u003e 1$. Then $N \\ge 1$.\n12. For $n \\ge N$, we get $d_X(x_n, a) \u003c 1/n \\le 1/N \u003c \\eta$. So $x_n \\to a$.\n13. But $d_Y(f(x_n), f(a)) \\ge \\varepsilon$ for every $n$. So no $N$ makes $d_Y(f(x_n), f(a)) \u003c \\varepsilon$ for all $n \\ge N$, and $f(x_n) \\not\\to f(a)$.\n14. So if $f(x_n) \\to f(x)$ for every convergent sequence $x_n \\to x$, then $f$ is continuous.\n\n## Idea\n\nContinuity in a metric space is the epsilon-delta condition. A convergent sequence eventually enters the $\\delta$-ball, so its image eventually enters the $\\varepsilon$-ball. If continuity fails at a point, the radii $1/n$ pick out bad points that converge to it while their images stay away.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "54E35"
    ],
    "name": "Proof that continuity is equivalent to sequential continuity in metric spaces",
    "proves": "0VGPKZ8MRK",
    "sources": [
      {
        "url": "https://proofwiki.org/wiki/Sequential_Continuity_is_Equivalent_to_Continuity_in_Metric_Space"
      }
    ],
    "tag": "YP4WMTVCPP",
    "uses": [
      "063W5JMY85",
      "0SY9ECQ19Q",
      "0YF82GBTND",
      "06JYPJ4WHH",
      "0A3GWZ86AY"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/YP4WMTVCPP/",
  "links": [
    {
      "field": "proves",
      "target": "0VGPKZ8MRK"
    },
    {
      "field": "uses",
      "target": "063W5JMY85"
    },
    {
      "field": "uses",
      "target": "0SY9ECQ19Q"
    },
    {
      "field": "uses",
      "target": "0YF82GBTND"
    },
    {
      "field": "uses",
      "target": "06JYPJ4WHH"
    },
    {
      "field": "uses",
      "target": "0A3GWZ86AY"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/YP4WMTVCPP.md",
  "msc": [
    "54E35"
  ],
  "name": "Proof that continuity is equivalent to sequential continuity in metric spaces",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YP4/YP4WMTVCPP.md",
  "status": null,
  "tag": "YP4WMTVCPP",
  "uri": "m2://YP4WMTVCPP",
  "url": "https://math2.tamnd.com/wiki/proof-that-continuity-is-equivalent-to-sequential-continuity-in-metric-spaces/",
  "writing": {
    "tag": "YP4WMTVCPP",
    "kind": "proof",
    "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
      }
    ]
  }
}