I have to proove that given $X$ a normed space and $Y$ a Banach space, if the sequence of bounden linear operators from $X$ to $Y$ $\{A_n\} \rightarrow A$ and the sequence $\{x_n\} \rightarrow x$ then $A_n x_n \rightarrow Ax$ in $Y$.
So, showing that $||A_n x_n -Ax|| \rightarrow 0$ when $n \rightarrow \infty$ is enough, but I don´t know where to apply the convergence of $x_n$, I also don´t know if I should use the definition of operator norm $||A||= min\{c\ge 0 : ||Av|| \le c||v||\}$
I tried to write the norm as $||A_n x_n - Ax||= ||A_n (x_n -x +x)-A(x)||$ and that doesn´t seem to take me anywhere.
Thanks for your time.