This is a common theorem and is proven in many books. I am confused with a particular part of the proof. This image has been taken from Christopher Heils notes.
If $A_n$ is a cauchy sequence in $B(X,Y)$ then we know that $||An-Am||\rightarrow 0$ as $n,m\rightarrow 0$. It then follows that for any $f\in X$ it must be that $||A_nf-A_mf||\rightarrow 0$. But I do not understand how we get $||A_nf-A_mf||\leq||A_n-A_m|| \,||f||$.