I have a question which appeard during my modifications of Weiestrass theorem.
Let $X$ be a reflexive Banach space. I have a set $A$ which is closed and convex, so from Mazur lemma it is weakly closed. Also $A$ is bounded and $A\subset B(0,r)$. Since $X$ is reflexive, then unit ball is weakly sequentially compact, so also $B(0,r)$ is weakly sequentially compact. My question is as follows: is $A$ also weakly sequentially compact? If so, do you know where I can find the proof?