How to derive Riemann-Lebesgue lemma from Bessel inequility? I encountered Riemann-Lebesgue Lemma in Functional Analysis. It can be viewed as a corrollary of Bessel inequtility in the following picture: ![enter image description here]( However, I can't see why it is ' **in particular** ' ? Can anyone give me some hint? My brain just short-circuit ...

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