Let S be a nonempty, closed, convex set in a Euclidean space X and y a point

Question:

Let S be a nonempty, closed, convex set in a Euclidean space X and y a point outside S (figure 3.4). Show that
1. There exists a point x0 e S which is closest to y, that is,
||x0 - y|| ≤ ||x - y|| for every x ∊ S
2. x0 is unique
3. (x0 - y)T(x - x0) ≥ 0 for every x ∊ S
Finite dimensionality is not essential to the preceding result, although completeness is required.