Let $A \subset \mathbb{R^n}$ be a closed subset and $x \in \mathbb{R^n}$.
How to prove that a point $p \in A$ exists such that $d_2(p,x)=\inf\limits_{q \in A}d_2(q,x)$?
I tried to focus upon the case that $A$ is a bounded set and compact. I used:
$q \in \mathbb{R^n} \Leftrightarrow d(q,\mathbb{R^n})=0$. So it's $q=p$.
But I don't know how to use the infimum here.
Does it work or is there another way to prove this?