About the existence of a convergent sequence

Let $(A_n)$ be a set sequence in a Banach space wheresuch that $A_n$ is nonempty, closed and convex for every $n=1,2\dots$. Assume that $\displaystyle\lim_{n,m\to \infty} d(A_n,A_m)=0$ where d is the Hausdorff distance between two sets.

My question is whether there exists a convergent sequence $(x_n)$ satisfying $x_n\in A_n$ for every $n$?

• Great question! Can it be shown that the answer is no if we weaken the assumptions to: 1) the base space $X$ is just a complete metric space, and 2) the sets $A_n$ are nonempty and closed (but not necessarily convex). – Dominic van der Zypen Jul 4 '17 at 6:26