An Injective Self-Map of a Finite Set is a Bijection

Let $\mathbb{N}$ be the set of natural numbers with successor map $S$ as in that definition, let $n\in\mathbb{N}$, and let $[n]$ be the initial segment determined by $n$. Call a map $u:X\to Y$ between sets injective if $u(x)=u(x')$ implies $x=x'$ for all $x,x'\in X$. The notions finite and has $n$ elements are those of the indicated definitions.

Then the following hold.

1. (Initial segments) Every injective map $u:[n]\to[n]$ is a bijection, and hence a permutation of $[n]$.

2. (Finite sets) If $X$ is a nonempty finite set and $u:X\to X$ is injective, then $u$ is a bijection from $X$ onto $X$.