# Nested sequence of sets in Hilbert space

How can I prove that nested sequence of non-empty bounded closed convex sets in Hilbert space have nonempty intersection?

I just don't know where to start.

Thanks

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Do you know the weak topology for Hilbert space? – GEdgar Nov 6 '11 at 21:03
I don't know what it is. – Slon Nov 6 '11 at 21:18
What do you know about closed bounded convex sets in a Hilbert space, then? – t.b. Nov 6 '11 at 21:48

## 1 Answer

This is Cantors Intersection Theorem. The (simple) proof can be found here for example. You need to use the fact that closed bounded convex subsets in a Hilbert space H are weakly compact.

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Unfortunately, if he doesn't know what is the weak topology, he probably also doesn't know anything about "weakly compact". This illustrates the standard wisdom: For problems with homework, seek help from the instructor. – GEdgar Nov 7 '11 at 1:40