If it is given that $X$ is a reflexive Banach space. Let $K \subset X$ be a norm closed and norm bounded convex set. I want to show that $K$ is weakly compact. I have the following idea but I am not sure about how to prove one thing.
Idea for proof: $K$ is bounded therefore there exists some closed ball $B_{r}$(of radius $r$) such that $K \subset B_{r}$. We know that closed ball $B_{r} = rB_{1}$, therefore the closed unit ball $B_{1} = \frac{1}{r}B_{r}$. By Kakutani's Theorem $B_{1}$ is weakly compact. I'm not sure then if we can state that since $B_{1}$ is weakly compact it follows that $B_{r}$ is weakly compact. If it does then since $K$ is weakly closed(since it is norm closed) it follows that $K$ is also weakly compact.
So basically I want to know if it follows that if $B_{1}$ is weakly closed then any closed ball $B_{r}$ is weakly compact? If so, could I get a hint of how to show this? If not are there any suggestions regarding how to prove that $K \subset X$ is weakly compact?