How to derive Riemann-Lebesgue lemma from Bessel inequility?

Question

I encountered Riemann-Lebesgue Lemma in Functional Analysis. It can be viewed as a corrollary of Bessel inequtility in the following picture:

However, I can't see why it is 'in particular' ? Can anyone give me some hint? My brain just short-circuit ...

Answer 1

The series $\sum_{k=1}^\infty|\left<x,x_k\right>|^2$ is convergent so its terms tend to zero, that is $\lim_{k\to\infty}|\left<x,x_k\right>|^2=0$. Therefore $\lim_{k\to\infty}\left<x,x_k\right>=0$