If $x_n\rightarrow x$ and $y_n\rightarrow y$ in a Hilbert space then $\left<x_n,y_n\right>\rightarrow\left<x,y\right>$ as $n\rightarrow\infty$.
So the way I started: using Cauchy-Schwarz: $$ |\left<x_n,y_n\right>-\left<x,y\right>|=|\left<x_n,y_n\right>-\left<x_n,y\right>+\left<x_n,y\right>-\left<x,y\right>|\leq|\left<x_n,y_n-y\right>|+|\left<x_n-x,y\right>|=0 $$ as $n\rightarrow\infty$. Is this correct?