Statements about convergence in algebra operators

Question

Let $e_i = {\delta_{ij}} \in l_2$
$A_n, B_n \in \mathcal{L}(l_2)$
$A_n: {x_i} \rightarrow x_ne_1$
$B_n: {x_i} \rightarrow x_1e_n$
$\Rightarrow A_n \rightarrow 0; B_n \rightarrow 0$
$\not\exists s - lim B_n$
$A_nB_n \rightharpoonup 0$
Where s - strong operator topology. I think, that we can prove this for $\mathcal{L}(l_\infty)$. Can we reduce our $l_2$ case to this situation?