An easy version of Riemann-Lebesgue Lemma on $L^2([-\pi,\pi])$

Question

Question

Let $\left\{{\varepsilon}_{k}\right\}$ be an orthonormal set (it may be complete or not) in a Hilbert space $H$. Explain why for any $x \in H^{ \perp}$ $$ \lim _{k \rightarrow \infty}\left\langle x, {\varepsilon}_{k}\right\rangle=0 $$ Let $f \in L^{2}([-\pi, \pi]).$ Using the above fact show the following: $$ \lim _{n \rightarrow \infty} \int_{-\pi}^{\pi} f(x) \cos (n x )dx=\lim _{n \rightarrow \infty} \int_{-\pi}^{\pi} f(x) \sin (n x)dx=0 $$

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I know that this is a rather easy version of Riemann-Lebesgue Lemma. However, I couldn't simply understand how to use the above fact. I'm open for any hint about reasoning to begin proving the claim.

Answer 1

You always have Bessel's inequality: $$ \sum_{n}|\langle x,e_n\rangle|^2 \le \|x\|^2. $$ Therefore, you have a convergent sum, and the general term of a convergent sum always converges to $0$.