Studying for topological vector spaces I came across this problem:
Let $E$ be a normed space and let $K$ be a weakly compact subset of $E$. Then $K$ is sequentially compact, norm bounded and proximal.
We know that the dual space is a Banach space so from the weak compactness we can conclude that for every $ f \in E^*$ $$ \sup\{|f(x)|,x\in K\} < \infty$$ and from the uniform boundedness principle we get that $K$ is norm bounded. Now assuming that there exists a sequence $x_n$ with no convergent subsequence we fix $x\in K$ , $r>0$ and we consider the sets $$\{n\in \mathbb N:\|x_n-x\|>r\}$$ which have to be infinite. Doesn't this contradict the fact that $x_n$ is bounded? What about $K$ being proximal?