Strong induction

Base case. $n=2$ is prime, so trivially a product of primes.

Inductive step. Suppose all integers $2\le k<n$ have prime factorization. For $n$:
- If $n$ is prime, done.
- If $n$ is composite, $n=ab$ with $2\le a,b<n$. By strong induction, $a$ and $b$ are products of primes, so $n=ab$ is too.

Conclusion. Every integer $\ge2$ is a product of primes.