Converse statement of Mazur's lemma

Question

I want to prove that the following converse statement of Mazur's lemma is true.

Let $(x_n)$ be a bounded sequence in the Banach space X, then $(x_n)$ is weakly null if for all subsequences $(z_n)$ of $(x_n)$ and $\epsilon>0$, there is a convex combination $z=\sum^k_{j=1}\lambda_j z_j$ of $(z_n)$ such that $\|z\|\le\epsilon.$

I know we should use the fact that the weak and norm closures of a convex set coincide with each other. But how to use the condition that the sequence is bounded?

Thanks in advance!

Answer 1

Hint: Assume that $(x_n)$ does not converge weakly towards zero. Then, (by invoking just the definition) there is $x^* \in X^*$, $\epsilon > 0$ and a subsequence $(z_n)$ of $(x_n)$, such that $|x^*(z_n)| > \varepsilon$ for all $n$. Can you continue?