Induction from an arbitrary base case

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 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.