Given $X$ and $Y$ are disjoint compact subsets of a metric space $M$ prove that there exists $x_0 \in X$ and $y_0 \in Y$ such that for all $x \in X$ and $y \in Y$ we have $d(x_0,y_0)\leq d(x,y)$.
How would one go about this? and are the $x_0,y_0$ values unique?