Uniquness and Exisstence of One Theorem

Question

I need a short and nice Proof for Uniqueness and Existence of the following theorem:

Suppose (H, <0,0> ) is a Hilbert space, and M is a closed convex set and $x \in H$, then there is a unique element $y_x \in M$ such that:

$ || x-y_x|| = inf $ { $ || x-t || : t \in M $ }

could anyone help me?

i have some more complicated proof, but i need a short and complete proof :).

Answer 1

Hint:

Existence : Without loss of generality we can suppose x=0 (we could simply translate by x the set A).

Uniqueness : Suppose there were $a_0,b_0∈A$ such that ||$a_0$∥=∥$b_0$∥=d and using the parallelogram law.