TITLE: Prove a closed set is a countable union of compact sets, in $\mathbb{R}^n$.
QUESTION [6 upvotes]: 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, :)

REPLY [8 votes]: 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
$$
C=\bigcup_{\substack{n\in\mathbb{N}\\n>0}}C_n
$$