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Let $X$ be a Banach space and let $(x_n)_{n=1}^\infty$ be a weakly convergent sequence in $X$ to some $x$. I want to prove that there exists a sequence $(z_n)_{n=1}^\infty$ such that $z_n\in {\rm conv}\{x_h|h\leqslant n\}$ and it converges strongly to $x$. I can prove it for $z_n\in {\rm conv}\{x_h|h\geqslant n\}$ using the fact that a convex subset is closed only and only it's weakly closed, but I don't know how to deal with this case.

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See theorem 3.13 in Rudin's Functional analysis.

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