Compact Hausdorff space and closure

Question

Let $X$ be a compact Hausdorff space, let $A \subset X$ and let $U$ is open subset of $X$ such that $\overline A \subset U$. prove that there exist an open set $W$ such that $\overline A \subset W$ and $\overline W \subset U$.

Answer 1

$X$ is normal and $\overline{A}$ is closed. QED.