 Weak convergence of bounded disjointly supported sequences in lp

Let $1<p<\infty$. Every norm-bounded sequence in ${\ell }^p$ whose terms have pairwise disjoint supports converges weakly to zero. Indeed, the average of $N$ distinct terms has norm $O(N^{1/p-1})$, so Mazur theorem gives weak convergence through its convex-combination criterion.