Existence of points 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.

We can assume that $X$ and $Y$ are disjoint. Suppose that $Y$ is bounded. Then, both $X$  and $Y$ are compact, and the result is obvious: Take a sequence $(x_n, y_n)$ in $XxY$ such that $d(x_n,y_n)$ converges to $D=d(X,Y).$ Then, $x_0=\lim x_n$ and $y_0 = \lim y_n.$
Now consider the original question.  Let $C\in X$ satisfy $d(C,Y)\le 2D$ and let $Y_1=Y\cap B,$ where $B$ is the ball of center $C$ and radius $4D.$  Clearly, $d(X,Y)=d(X,Y_1).$  Since $Y_1$ is closed and bounded, we have reduced to the case solved in the first paragraph.
