---
tag: 0GF6AJEBZ0
kind: statement
role: theorem
name: Induction from an arbitrary base case
msc: [03F30]
uses: [0ASJVEFA9J, 0Y9H52BT03]
sources:
  - url: https://en.wikipedia.org/wiki/Mathematical_induction
    locator: Base case other than 0 or 1
---

## Statement

Let $m$ be a natural number and let $P(n)$ be a property of natural numbers $n$ with $n \ge m$.
Suppose that $P(m)$ holds.
Suppose also that for each natural number $n \ge m$, $P(n)$ implies $P(n + 1)$.
Then $P(n)$ holds for every natural number $n \ge m$.

## Remarks

The proof applies ordinary [[induction|0ASJVEFA9J]] to the property $Q(k)$: "$P(m + k)$ holds".
For example, induction from $m = 4$ shows that $2^n \ge n^2$ for every natural number $n \ge 4$.

## Explanations

### L1 human

Sometimes a property is false for small numbers and true from some point on.
Then you start the induction at that point instead of at $0$.
The dominoes start falling at domino number $m$, and the dominoes before it do not matter.