Prove that every weakly convergent sequence in $l^1$ converges.
By Riesz on $L^p$ spaces, every linear functional $L\in (l^1 )^*$, is $L(x) = \langle u,x\rangle$ for some $u\in l^{\infty}$ (which is to say that $|u_i|\leq C$ for some $C$ and all $i\geq 1$ (here $x$ and $u$ are sequences of complex numbers).
In particular, we can easily deduce that if $x^i \rightarrow x$ weakly then $x^i_j \rightarrow x_j$ for every $j$, but we really need to show that $\sum_0^{\infty}|x^i_j-x_j| \rightarrow 0$ as $i\rightarrow \infty$, which requires more.
I'm really not sure how to proceed with this. Hints?
