Given a metric space $(X,d)$. Let $A$ and $B$ be two nonempty subsets of $X$. If $A$ is compact and $B$ is closed, then there exist $x_0\in A$ and $y_0\in B$, such that $d(A,B)=d(x_0,y_0)$?
I know that, if $A$ and $B$ are both compact, then the conclusion is right. But if $B$ is just a closed subset?
Thank you for any help.
