I worked in this problem too:
But after that I could not think of anything that can help me about the $l_p$ case. I mean, $\{ n^{1/p} e_n \}$ has $0$ as a weak accumulation point but no subsequence of this set weakly converges to $0$ (with $p \in [1,\infty)$) ?
I tried to use something about my $l_2$ proof but it uses strongly facts about Hilbert spaces.
For example, to prove that $0$ is an accumulation point. Given a weak neighborhood $W$ of $0$ and a natural $n_0$ if I suppose that, for all $n \geq n_0$, is true that $\sqrt{n}e_n \notin W$ I could find an absurd using the definition of $W$. But I used the fact that $\{ e_n \}$ is a hilbert basis.
So, do you have any hint about the $l_p$ case?
obs: I'm still studying english, sorry about my errors =p
Thanks!