{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\nWe apply ordinary [[induction|0ASJVEFA9J]] to a shifted property.\nLet $m$ and $P$ be as in the statement.\nFor each natural number $k$, let $Q(k)$ be the property \"$P(m + k)$ holds\".\nHere $+$ is the [[addition|0W5DSDFQ4G]] of natural numbers.\n\n1. For each natural number $k$, we have $m \\le m + k$ by the definition of the [[order|0Y9H52BT03]]. So $P(m + k)$ makes sense.\n2. Base case: $m + 0 = m$, and $P(m)$ holds by hypothesis. So $Q(0)$ holds.\n3. Induction step: suppose that $Q(k)$ holds, that is, $P(m + k)$ holds.\n4. By step 1, $m + k \\ge m$. So the second hypothesis of the statement, applied to $n = m + k$, gives $P((m + k) + 1)$.\n5. By the [[associativity of addition|0C32J27WQC]], $(m + k) + 1 = m + (k + 1)$. So $Q(k + 1)$ holds.\n6. By induction, $Q(k)$ holds for every natural number $k$.\n7. Now let $n$ be a natural number with $n \\ge m$. By the definition of the order, there is a natural number $k$ with $m + k = n$.\n8. By step 6, $Q(k)$ holds, that is, $P(n)$ holds. This holds for every $n \\ge m$.\n\n## Idea\n\nShift the numbering so that the first domino gets the number $0$.\nThe number $m + k$ becomes the number $k$, and ordinary induction applies to the shifted property.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "methods": [
      "0ASJVEFA9J"
    ],
    "msc": [
      "03F30"
    ],
    "name": "Proof of induction from an arbitrary base case",
    "proves": "0GF6AJEBZ0",
    "sources": [
      {
        "locator": "Base case other than 0 or 1",
        "url": "https://en.wikipedia.org/wiki/Mathematical_induction"
      }
    ],
    "tag": "CDCZ3DZ40E",
    "uses": [
      "0ASJVEFA9J",
      "0Y9H52BT03",
      "0W5DSDFQ4G",
      "0C32J27WQC"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/CDCZ3DZ40E/",
  "links": [
    {
      "field": "proves",
      "target": "0GF6AJEBZ0"
    },
    {
      "field": "uses",
      "target": "0ASJVEFA9J"
    },
    {
      "field": "uses",
      "target": "0Y9H52BT03"
    },
    {
      "field": "uses",
      "target": "0W5DSDFQ4G"
    },
    {
      "field": "uses",
      "target": "0C32J27WQC"
    },
    {
      "field": "methods",
      "target": "0ASJVEFA9J"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/CDCZ3DZ40E.md",
  "msc": [
    "03F30"
  ],
  "name": "Proof of induction from an arbitrary base case",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CDC/CDCZ3DZ40E.md",
  "status": null,
  "tag": "CDCZ3DZ40E",
  "uri": "m2://CDCZ3DZ40E",
  "url": "https://math2.tamnd.com/wiki/proof-of-induction-from-an-arbitrary-base-case/",
  "writing": {
    "tag": "CDCZ3DZ40E",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}