Mazur lemma

If $x_n$ converges weakly to $x$ in a Banach space, there are convex combinations of each tail $\{x_n,x_{n+1},\dots \}$ that converge in norm to $x$.

The weak closure and norm closure of a convex subset of a real or complex normed space coincide. If a point lies outside the norm closure, the Hahn-Banach separation theorem supplies a weakly continuous linear functional that separates it from the convex set.