Prove a closed set is a countable union of compact sets, in $\mathbb{R}^n$.

Question

I'm studying Sard's theorem and I want to know why is true that, in $\mathbb{R}^n$, every closed subset can be expressed as a countable union of compact sets. Thank you, :)

Answer 1

Consider $\bar{B}(0,n)$, the closed ball with center $0$ and radius $N$. If $C$ is a closed set, consider $$ C_n=C\cap\bar{B}(0,n) $$ Then $C_n$ is a closed subset of $\bar{B}(0,n)$, which is compact, hence $C_n$ is compact. Obviously $$ C=\bigcup_{\substack{n\in\mathbb{N}\\n>0}}C_n $$