IMC 2003 Problem 9

Let $A$ be a closed subset of $\mathbb{R}^{n}$ and let $B$ be the set of all those points $b \in \mathbb{R}^{n}$ for which there exists exactly one point $a_{0}\in A$ such that $|a_{0}-b|= \inf_{a\in A}|a-b|$ . Prove that $B$ is dense in $\mathbb{R}^{n}$ ; that is, the closure of $B$ is $\mathbb{R}^{n}$