countable intersection of closed convex bounded subsets reflexive banach space is non empty. 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?
Hint: Do you know the Eberlein-Shmulyan theorem?
the proof can be seen in Introduction to Banach Space Theory by Robert E. Megginson.