Suppose $T: H \rightarrow H$ is a compact operator, $H$ is a Hilbert space, and let $(A_n)$ be a sequence of bounded linear operators on $H$ converging strongly to $A$. Show that $A_nT$ converges in operator norm to $AT$.
I'm attempting to prove this by contradiction: That is, there exists some $\epsilon > 0$ such that $ \|A_nT-AT\| = \sup_{\|f\|=1}\|(A_nT-AT)f\| > \epsilon$ for all $n$. How can I exploit the compactness of $T$?