Let $A$ be a compact set and $B$ a closed set ($\varnothing\ne A,B\subseteq \mathbb{R}^n$). Prove there's a minimum distance between $A$ and $B$.
In class we've seen that there's a minimum distance between a compact set $A$, and a point $x_0\notin A$. I thought about utilizing it as a generalization.
First we may assume the points (if exist) must be on the spheres of the sets. For each $x_0$ in the sphere of $B$ there's a point $y_0$ in the sphere of $A$ such that $\forall y\in A: \|y_0-x_0\| \le \|y-x_0\|$.
So we define $f:A\to \mathbb{R}$ such that $f(x) = \text{minimumDistance(x,B)}$.
Is that a good start? How should I proceed?