= 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> gives
$$
\|z-y\|^2=\frac12\|x_1-y\|^2+\frac12\|x_2-y\|^2-\frac14\|x_1-x_2\|^2
=d^2-\frac14\|x_1-x_2\|^2<d^2,
$$
contradicting minimality. Therefore \b[the nearest point is unique]. Together with the preceding existence proof, this defines the metric projection onto a nonempty closed <convex set>.
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