 Mazur's 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 $x$.

Mazur's Lemma is a result in functional analysis, specifically in the context of convex analysis and the study of weak compactness in Banach spaces. The lemma is named after the Polish mathematician Stanisław Mazur. The central idea of Mazur's Lemma concerns weakly convergent sequences and the nature of the weak closure of convex sets.