{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\n1. The inequality $\\norm{x + y} \\le \\norm{x} + \\norm{y}$ is axiom 1 in the definition of a [[norm|0V2WP9B1MF]].\n2. Write $x = (x - y) + y$. By axiom 1, $\\norm{x} \\le \\norm{x - y} + \\norm{y}$. So $\\norm{x} - \\norm{y} \\le \\norm{x - y}$.\n3. In the same way, $y = (y - x) + x$ gives $\\norm{y} - \\norm{x} \\le \\norm{y - x}$.\n4. Since $y - x = (-1)(x - y)$, axiom 2 gives $\\norm{y - x} = \\abs{-1} \\norm{x - y}$. The [[absolute value|0KW2X0TFBX]] of $-1$ is $1$, so $\\norm{y - x} = \\norm{x - y}$.\n5. Put $t = \\norm{x} - \\norm{y}$, a real number. By steps 2 to 4, both $t \\le \\norm{x - y}$ and $-t \\le \\norm{x - y}$.\n6. The absolute value $\\abs{t}$ equals $t$ or $-t$. So step 5 gives $\\bigl| \\norm{x} - \\norm{y} \\bigr| \\le \\norm{x - y}$.\n\n## Idea\n\nThe first inequality is an axiom of a norm. For the reverse form, apply the axiom to $x = (x - y) + y$ and to $y = (y - x) + x$. Since $\\norm{y - x} = \\norm{x - y}$, the two bounds together control $\\abs{\\norm{x} - \\norm{y}}$.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "46B20"
    ],
    "name": "Proof of the triangle inequality for norms",
    "proves": "0FJZMBX0ZZ",
    "sources": [
      {
        "locator": "Normed vector space",
        "url": "https://en.wikipedia.org/wiki/Triangle_inequality"
      }
    ],
    "tag": "5XAGK7S3WR",
    "uses": [
      "0V2WP9B1MF",
      "0KW2X0TFBX"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/5XAGK7S3WR/",
  "links": [
    {
      "field": "proves",
      "target": "0FJZMBX0ZZ"
    },
    {
      "field": "uses",
      "target": "0V2WP9B1MF"
    },
    {
      "field": "uses",
      "target": "0KW2X0TFBX"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/5XAGK7S3WR.md",
  "msc": [
    "46B20"
  ],
  "name": "Proof of the triangle inequality for norms",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/5XA/5XAGK7S3WR.md",
  "status": null,
  "tag": "5XAGK7S3WR",
  "uri": "m2://5XAGK7S3WR",
  "url": "https://math2.tamnd.com/wiki/proof-of-the-triangle-inequality-for-norms/",
  "writing": {
    "tag": "5XAGK7S3WR",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}