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

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!

Yes, it holds, because the map $$(x,y)\mapsto\langle x,y\rangle$$ is continuous and continuous maps map convergent sequences into convergent sequences.
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.