Let $\{x^k\}$ be a weak convergent sequence in $\ell_1$, and its weak limit is 0. Is the following property true:
For $\forall \epsilon >0$ and $\forall n>0$, there exists a K, s.t. $\sum_{i=1}^{n}|x^K_i|<\epsilon.$
This comes from a proof I read that tries to prove the equivalence of weak and strong convergence in $\ell_1$. It uses this property which I don't know why.