Let $H$ be a Hilbert Space. Let $\{A_n\}$ be a sequence of bounded operators in $H$, and $A\in B(H)$.
If $\|A_nf - Af\|\to 0$ uniformly for $f\in H_{\|.\|=1}\ $, prove that $\|A_n - A\|\to 0$.
Using uniformly boundedness principle, I know $\{A_n-A\}$ is uniformly bounded, but I do not know how to prove convergence. Please help me. Thanks.
EDIT: Is the following argument correct? For $\epsilon>0$, there is $h\in H$ such that $\|A_n-A\|-\epsilon < \|A_nh-Ah\|$. Now when $n\to \infty$, we have desired result.