Let $(M,d)$ be a compact metric space. Suppose that $(F_n)$ is a decreasing sequence of nonempty closed sets in $M$, and that $\bigcap_{n=1}^\infty F_n$ is contained in some open set $G$. Then $F_n \subset G$ for all but finitely many $n$.
I know that $\bigcap_{n=1}^\infty F_n \neq \emptyset$, but I'm having trouble proving the above statement.
Suppose not. Then there is an infinite sequence of $F_n$ such that all the $F_n$ aren't contained inside $G$. Since $G$ is open, each $F_n \setminus G$ is also closed; each is also nonempty by assumption. Consider the intersection of those.