Suppose $C$ is a compact operator and $M_n$ is a sequence of bounded linear operators converging pointwise to another bounded linear operator $M$. Show that $||CM_n - CM|| \rightarrow 0$.
I know how to prove $||M_n C - MC|| \rightarrow 0$ (using Ascoli-Arzela theorem) but get stuck after changing the order of $M_n$ and $C$. Can anyone shed some light? All hints will be appreciated!