Consider the Mazur's Lemma (H. Brezis - "Functional analysis, ..."):
"Assume $(x_n)$ converges weakly to $x$. Then there exists a sequence $(y_n)$ made up of convex combinations of the $x_n$'s that converges strongly to $x$.''
The Lemma says that "there exists a sequence...''.
Is it true that $\textbf{every}$ sequence $(y_n)$ made up of convex combinations of the $x_n$'s converges strongly to $x$?
In particular, if we consider $$y_n = \frac{1}{n}(x_1 + x_2 + ... +x_n),$$
is it true that $y_n$ converges strongly to $x$?