Continuity of scalar product

In a Hilbert space $H$ with inner product and associated norm, why would if $\|x-x_n\| \longrightarrow 0$ and $\|y-y_n\| \longrightarrow 0$ also $\langle x_n,y_n\rangle \longrightarrow\langle x,y\rangle$?

I understand that by Cauchy-Schwarz $\lvert\langle x-x_n,y-y_n\rangle\rvert \leq \|x-x_n\|\cdot\|y-y_n\|\xrightarrow{n\to\infty} 0$ but how do I get to $\lvert\langle x,y\rangle-\langle x_n,y_n\rangle \rvert\longrightarrow 0$?

Solution 1:

Hint:

$$|\langle x,y\rangle-\langle x_n,y_n\rangle|=|\langle x,y\rangle-\langle x_n,y\rangle+\langle x_n,y\rangle-\langle x_n,y_n\rangle|;$$

Grouping the terms, using the triangular inequality for $|\cdot|$ and Cauchy-Schwarz helps.