Strong induction

Statement

Let $P(n)$ be a property of natural numbers $n$. Suppose that for each natural number $n$, if $P(k)$ holds for every natural number $k < n$, then $P(n)$ holds. Then $P(n)$ holds for every natural number $n$.

Remarks

The hypothesis for $n = 0$ asks for a proof of $P(0)$ with no assumption, because no natural number is less than $0$. The proof applies ordinary induction to the property "$P(k)$ holds for every $k < n$". A standard use is the proof that each integer $n \ge 2$ is a product of primes.

Explanations

L1 human

In ordinary induction, you may use only the case just before $n$. In strong induction, you may use all the cases before $n$. This helps when $n$ breaks into smaller pieces that are not just $n - 1$, for example when $n = a b$ with $a$ and $b$ smaller than $n$.