# real analysis – Prove inequality in metric spaces

We prove that the statement: Let $$(mathbb{R}^n,d)$$ is a
euclidean space, $$X$$ is a non-empty closed subset of $$mathbb{R^n}$$. If $$xin mathbb{R}^n-X$$, then exist $$yin X$$ such that for all $$zin X$$, $$d(x,y)leq d(x,z)$$.

I thought going through a contradiction, it is not easy for me.
On the other hand, i tried to find the existence, but
i failed it. I am interested in this problem, can you give me a hint?