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Nested sequence of sets in Hilbert space

$\{A_n\}$ is a descending sequence of closed convex bounded subsets in Hilbert space. Why can't the intersection be empty?
I'm not sure how to start proving that, I will be glad to get a hint. I think the following theorem might help: In Hilbert space every non empty, closed, convex subset contains a unique element of smallest norm.

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  • I don't think there's any possible about it. – JSchlather Nov 09 '11 at 22:36
  • @JacobSchlather: In case you didn't know, the "possible" (and everything else in t.b.'s comment) is part of the automatically generated message that the software posts when someone votes to close as a duplicate of another question. It appeared because t.b. was first to vote to close (and I guess they include "possible" because it should also apply to less clear-cut cases). – Jonas Meyer Nov 09 '11 at 22:52
  • @Jonas, I did not know that. Thanks. – JSchlather Nov 10 '11 at 02:08

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