Here is part of the question in my HW. Let$\ \{C_n\}\subset X $ be a bounded nested decreasing sequence of closed and convex sets. I am asked to show that $\bigcap C_n \not= \emptyset$ iff X is reflexive. My intuitive tell me that Kakutani theorem can possibly be used to find a convergent subsequence in each $C_n$. However,I can't show that these sequences converge to a same point. Alternatively, The hint says that the sets $C_n$ are weakly closed by using the Separating Hyperplane theorem and its corollaries. If I can show the weakly closed part , then by weakly compactness there might be an accumulation point in the intersection.
Could anyone tell me how to show the sequence is weakly closed as suggested in the hint and the nonempty intersection part? Thanks in advance !