Let $A \subseteq \mathbb{R}^n$ be a closed proper subset. For $x, y \in \mathbb{R}^n$, denote the usual (Euclidean) distance between them by $d(x, y)$. Let $x \in \mathbb{R}^n \setminus A$; define $\delta := \inf\{ d(x, y) \mid y \in A \}$. Show that there exists $y \in A$ such that $\delta = d(x, y)$.