Show that minimum distance to convex set equals maximum distance to hyperplane

Question

If $C \subset \mathbb{R}^n$ s a nonempty closed convex set, how do we show that for every point $u \in \mathbb{R}^n \setminus C$ the minimum distance from $u$ to any point in $C$ equals the maximum distance from $u$ to a hyperplane separating $u$ from $C$?

I kind of get the geometric intuition, but how do I show this rigorously?