[Math] Convergence of the inner product in Hilbert Space

Question

I'm starting to study Hilbert Spaces for the very first time in my life and I had difficulty to understand one very simple proof:

Let $\{x_n:n=1,2,…\}$ be a sequence of vectors in the space; the sequence is said to converge to an element $x$ (of the space) $x_n\rightarrow x$ if $\lim_{n\rightarrow\infty}||x_n-x||=0$.

And then it's also stated that if $x_n\rightarrow x$ then $\langle x_n,z\rangle\rightarrow \langle x,z\rangle$.

I'm a little confused, first the definition of convergence implies the use of the norm, so I guess that I should take absolute values instead for the inner product.

The book mentions the Schwarz Inequality as the tool needed to prove the statement, but honestly I don't see how could this be used in this context.
Any hint will be greatly appreciated.

Answer 1

Hint:

$$\langle x_n, z\rangle = \langle x_n - x + x, z\rangle = \langle x_n-x, z\rangle + \langle x, z\rangle$$

Now, use the Cauchy-Scwhartz inequality on $$\langle x_n-x, z\rangle.$$

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Note To refresh your memory, the inequality is $$\langle a,b\rangle \leq \|a\|\cdot\|b\|$$