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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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    @Jonas, I did not know that. Thanks.2011-11-10

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