 Solution

Suppose two distinct minimizers $x_1,x_2$ have the same minimum distance $d$ from $y$. Convexity puts their midpoint $z=(x_1+x_2)/2$ in $A$. Expanding the Euclidean norm givescontradicting minimality. Therefore the nearest point is unique. Together with the preceding existence proof, this defines the metric projection onto a nonempty closed convex set.