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?