Nested Sequence of Non-Empty Compact Sets 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.)

let $U \supseteq \bigcap_n K_n$ be open. Set $L_n = K_n \setminus U$, then the $L_n$ are a nested sequence of compact sets. Now $\bigcap_n L_n = \bigcap_n K_n \setminus U = \emptyset$, therefore there is an $m$ such that $L_n = \emptyset$ for $n> m$, which means that $K_n \subseteq U$ for these $n$.