WOT convergence of the operators Let $\\{A_n\\}$ and $\\{B_n\\}$ be sequences in $\mathfrak{B}(H)$ such that $A_n \to A$ in WOT and $B_k \to B$ in SOT. Show that $A_nB_n \to AB$ in WOT. This is from Conway's book "A Course in Funcional Analysis". Thank you for any hints.

Let $x,y \in H$. We have to show that $\langle A_n B_n x, y \rangle \to \langle A B x, y \rangle$. Try writing $$\langle A_n B_n x, y\rangle - \langle ABx, y\rangle = (\langle A_n B_n x,y\rangle - \langle A_n Bx,y\rangle) + (\langle A_n Bx,y\rangle - \langle ABx,y \rangle).$$ For the first parenthesized term, observe that $$|\langle A_n B_n x,y\rangle - \langle A_n Bx,y\rangle| \le \|A_n\| \|B_n x - Bx\| \|y\|.$$ Use the uniform boundedness principle to show $\sup_n \|A_n\| < \infty$ and conclude that this term goes to 0. The second term is easier.