Unit vector basis in $\ell_1$

Question

Can someone illuminate me with a hint about why it is the case that no subsequence of the unit vector basis $(e_n)$ of $\ell_1$ is weak Cauchy?

Answer 1

Let $(e_{n(k)})_k$ be any subsequence. Set $$\phi_{l} = \begin{cases} 1 & \text{if }l = n(2k) \text{ for some k} \\ 0 & \text{otherwise}\end{cases}$$otherwise. Then $\phi = (\phi_l)_{l\in \mathbb{N}} \in l_\infty = (l_1)^\ast$ and $\phi(e_{n(k+1)}) - \phi(e_{n(k)}) = \pm 1$, so $\phi(e_{n(k)}) \nrightarrow 0$ as $k \to \infty$.