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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 $n$?

I asked this question on MSE, but haven't got answers.

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  • $\begingroup$ Here is link to math.SE question: Existence of convergent sequences. $\endgroup$ – Martin Sleziak Jul 4 '17 at 6:23
  • $\begingroup$ 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). $\endgroup$ – Dominic van der Zypen Jul 4 '17 at 6:26
  • $\begingroup$ @DominicvanderZypen: Is your comment a question or a statement? It begins like a question but it doesn't end with a question mark, so how should we understand it? $\endgroup$ – Alex M. Jul 4 '17 at 8:42
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Fedor Petrov's answer to my more general question implies a positive answer to this question, as Banach spaces are complete metric spaces.

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