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If $X$ is a reflexive Banach space and $(C_n), n \in \mathbb{N}$ is a sequence of closed convex bounded sets with $C_{n+1}$ contained in $C_n$ for all $n \in \mathbb{N}$. How does one show that the countable intersection of $C_n$ for $n \in \mathbb{N}$ is not the empty set?

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    http://meta.stackexchange.com/questions/5234/how-does-accepting-an-answer-work2012-06-10

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Hint: Do you know the Eberlein-Shmulyan theorem?

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    If $x_n \in C_n$, Eberlein-Shmulyan says some subsequence has a weak limit point $x$. Show that $x$ is in all the $C_n$.2012-06-10
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the proof can be seen in Introduction to Banach Space Theory by Robert E. Megginson.

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    Please try to describe as much here as possible in order to make the answer self-contained.2013-04-30