Let $K_n$ be a nested sequence of non-empty compact sets in a Hausdorff space.
Prove that if an open set $U$ contains contains their (infinite) intersection, then there exists an integer $m$ such that $U$ contains $K_n$ for all $n>m$.
(I know that compact sets are closed in Hausdorff spaces. I can also prove that the infinite intersection of non-empty compact sets is non-empty, closed and compact in a Hausdorff space. I don't know how to use the fact that U is open.)