2.II.13B

Use the standard metric on $\mathbf{R}^{n}$ in this question.

(i) Let $A$ be a nonempty closed subset of $\mathbf{R}^{n}$ and $y$ a point in $\mathbf{R}^{n}$. Show that there is a point $x \in A$ which minimizes the distance to $y$, in the sense that $d(x, y) \leqslant d(a, y)$ for all $a \in A$.

(ii) Suppose that the set $A$ in part (i) is convex, meaning that $A$ contains the line segment between any two of its points. Show that point $x \in A$ described in part (i) is unique.

*Typos? Please submit corrections to this page on GitHub.*