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.

  • $\begingroup$ 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

Fedor Petrov's answer to my more general question implies a positive answer to this question, as Banach spaces are complete metric spaces.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.