The question: suppose $S$ is a set and $f(s): S \rightarrow \mathbb{N}_n$ is an injection. Then $|S|$ must be finite.
My proof: I will proceed by contradiction. Presume that $|S|$ is infinite. By the definition of an injection, every $s\in S$ must map to a unique $N\in \mathbb{N}_n$. By the pigeonhole principle, this is only possible if $|S|\le n$, which contradicts the presumption. Therefore $|S|$ must be finite.