Let $(e_k)$ be any orthonormal sequence in an inner product space $X$. Show that for any $x,y \in X$ $$\sum_{k=1}^\infty|\left<x,e_k\right>\left<y,e_k\right>|\le \|x\|\|y\|.$$
If $x=y$, then the proof becomes trivial, as then the result follows directly from Bessel's inequality $$\sum_{k=1}^\infty|\left<x,e_k\right>|^2\le \|x\|^2.$$
Now suppose that $x \neq y$, then \begin{align}\sum_{k=1}^\infty |\left<x,e_k\right>\left<y,e_k\right>| &= \sum_{k=1}^\infty \left|\left<\left<x,e_k\right>y,e_k\right>\right|\end{align} But I have no idea where to continue from here. Can anyone please help point me in the right direction?