Let $(X,||\cdot||)$ be a complete normed space. Let $F_1, F_2, F_3,\ldots\subseteq X$ be closed, non-empty subsets of $X$.
Assume that $F_1 \supseteq F_2\supseteq F_3\supseteq \cdots$
and that $\sup_{x,y\in F_n}||x-y||\to0$.
Show that the intersection of all $F_n$ is non-empty.
All I can possibly think to go on is that we need to show that there is some $f\in X$ such that $f\in F_1,F_2,F_3,\ldots$
Any pointers in the right direction?