IMC 2003 · Day 2 · P9

Let $b_0 \notin A$ (otherwise $b_0 \in A \subset B$), $\varrho = \inf\limits_{a \in A} |a - b_0|$. The intersection of the ball of radius $\varrho + 1$ with centre $b_0$ with set $A$ is compact and there exists $a_0 \in A$: $|a_0 - b_0| = \varrho$.

Denote by $B_r(a) = \{ x \in \mathbb{R}^n : |x - a| \le r \}$ and $\partial B_r(a) = \{ x \in \mathbb{R}^n : |x - a| = r \}$ the ball and the sphere of center $a$ and radius $r$, respectively.

If $a_0$ is not the unique nearest point then for any point $a$ on the open line segment $(a_0, b_0)$ we have $B_{|a - a_0|}(a) \subset B_{\varrho}(b_0)$ and $\partial B_{|a - a_0|}(a) \cap \partial B_{\varrho}(b_0) = \{a_0\}$, therefore $(a_0, b_0) \subset B$ and $b_0$ is an accumulation point of set $B$.