Be $X, Y$ two closed subspaces of $\mathbb{R}^n$ with $X$ bounded. Show the existence of points $x\_0 \in X, y\_0 \in Y$ such as $d(x\_0,y\_0) = d(X,Y)=\inf\_{x \in X, y \in Y} d(x,y)$.

My idea so far: I want to start by showing that $X\_r = \{ y \in \mathbb{R}^n | d(y,X) \leq r\}$ is compact. With this subspace, without losing generality, I would replace $Y$ by $Y \cap X\_r$ for $d(X,Y) \leq r$ where the intersection is nonempty.