Can someone verify the consistency of my work for the solution of this exercise?
Let $\{e_n\}$ be an orthonormal system in a linear space $X$ with inner product. Prove that, for any $x,y \in X$, $$ \sum_{n=1}^{\infty}|{\langle e_n,x \rangle}||{\langle{ e_n,y \rangle}}| \leq \|{x}\|\|{y}\|.$$
Here's my approach:Let $x=\sum_{n=1}^{\infty}\langle e_n,x \rangle e_n$ and $y=\sum_{m=1}^{\infty} \langle e_m,y \rangle e_m$. we have that:
$$|{\langle x, y \rangle}|= \bigg|{ \bigg\langle\sum_{n=1}^{\infty} \langle e_n,x \rangle e_n, \sum_{m=1}^{\infty}\langle e_m,y \rangle e_m \bigg\rangle}\bigg|\\ =\bigg|{\sum_{n=1}^{\infty}\sum_{m=1}^{\infty} \langle \langle e_n,x \rangle e_n, \langle e_m,y \rangle e_m \rangle}\bigg|\\ =\sum_{n=1}^{\infty}\sum_{m=1}^{\infty} |{\langle \langle e_n,x \rangle e_n, \langle e_m,y \rangle e_m \rangle}|\\ =\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}|{\overline{\langle e_n,x \rangle}{\langle e_m,y \rangle}{\langle e_n, e_m \rangle}}|\\ =\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}|{\overline{\langle e_n,x \rangle}{\langle e_n,y \rangle}\delta_{mn}}|\\ =\sum_{n=1}^{\infty}|{\overline{\langle e_n,x \rangle}{\langle e_n,y \rangle}}|\\ =\sum_{n=1}^{\infty}|{\langle e_n,x \rangle}||{\langle e_n,y \rangle }|$$
This looks right so far. Now by Cauchy-Schwarz inequality, $|\langle x,y \rangle | \leq \|x\|\|y\|$. But $|{\langle x, y \rangle}| = \sum_{n=1}^{\infty}|{\langle e_n,x \rangle}||{\langle e_n,y \rangle }| $. Therefore, $$\sum_{n=1}^{\infty}|{\langle e_n,x \rangle}||{\langle e_n,y \rangle }| \leq \|x\|\|y\|.$$
Is this enough to solve the problem? my professor said it doesn't solve it because by the other hand: $|\langle e_n,x \rangle| \leq \|x\|,\;|\langle e_n,y\rangle| \leq \|y\|$, so it's the case where $$\sum_{n=1}^{\infty}|{\langle e_n,x \rangle}||{\langle e_n,y \rangle }| \leq \sum_{n=1}^{\infty}\|x\|\|y\| = \infty$$.
For every $N$, the C.S inequality implies $$\sum_{n=1}^N|\langle e_n,x\rangle|\cdot |\langle e_n,y\rangle|\leq\left (\sum_{n=1}^N|\langle e_n,x\rangle|^2\right)^{1/2}\left (\sum_{n=1}^N|\langle e_n,y\rangle|^2\right)^{1/2}\leq \|x\|\cdot\|y\|$$ now let $N\to\infty$.