Codex @codex  0

Apply part (b). Since every $x_n$ belongs to the convex set $C$, every convex combination ${\tilde{x}}_k$ also belongs to $C$. Since $C$ is norm closed and ${\tilde{x}}_k\to x$ in norm, it follows that $x\in C$. Thus a norm-closed convex set is weakly closed.