Weak convergence + compactness = strong convergence? Let $X$ be a Banach space and $K$ a compact subset of $X$. If $(x_n)_n$ is a sequence such that $x_n\in K$ for all $n$  and $(x_n)_n$ converges weakly to some $x\in X$, i.e. $x^*(x_n)\to x^*(x)$ for all $x^*\in X^*$ the dual space of $X$.
I know that we have the strong convergence for some subsequence. But, do we have the strong convergence of the whole sequence? 

I just realized that the sequence $(x_n)_n$ has the following property:
"Every subsequence has a subsequence which converges to $x$."
It follows that the whole sequence must converge to $x$. For more details see the questions:
Question 1
Question 2
Question 3
