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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!

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

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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.

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