The question reads:
Let $U$ be an open subset in the compact metric space $X$. If $\{ F_{k} : k \in \mathbb{N} \}$ is a collection of closed subsets of $X$ such that $F_{k+1} \subset F_{k}$, for all $k \in \mathbb{N}$, and $\bigcap_{k=1}^{\infty} F_{k} \subset U$, then show that $F_{n} \subset U$, for some $n \in \mathbb{N}$.
I think I have a proof for this but it seems way too easy considering how much information I am given. I should add that my definition of compact means that every open covering has a finite subcovering.
$\cap_{k=1}^{\infty} F_{k} \subset U \Rightarrow \forall k \in \mathbb{N}, \exists \ x_{0} \in U$ such that $x_{0} \in F_{k} $
Let $F_{n} = \{x_{0}\}$, $n \in \mathbb{N}$
$F_{n} \subset U\ \ \ $ Q.E.D.