Injection from Finite Set to Itself is Surjection/Corollary

Theorem
Let $S$ be a finite set.

Let $f: S \to S$ be an injection.

Then $f$ is a permutation.

Proof
From Injection from Finite Set to Itself is Surjection, $f$ is a surjection.

As $f$ is thus both an injection and a surjection, $f$ is a bijection by definition.

Thus as $f$ is a bijection to itself, it is by definition a permutation.