Let $E$ be an inner product space and $\lim x_n = x$, $\lim y_n = y$. Is it true that $\lim \langle x_n, y_n \rangle = \langle x, y \rangle$

Question

Let $(E, \langle \cdot, \cdot \rangle)$ be an inner product space, and $(x_n),(y_n)$ sequences such that $\lim_{n \to \infty}x_n = x$, $\lim_{n \to \infty}y_n = y$ w.r.t to the induced norm. We define a sequence $(z_n)$ by $z_n := \langle x_n, y_n \rangle$.

My question: Is it true that $$\lim_{n \to \infty} z_n = \langle x, y \rangle$$

I guess that it holds, but I've not come up with a proof. Thank you for your help!

Answer 1

The answer is yes. Hint: note that $$|\langle x_n, y_n \rangle-\langle x, y \rangle| \leq |\langle x_n, y_n-y \rangle|+|\langle x_n-x, y\rangle|$$ and finally apply the Cauchy-Schwarz inequality.