Assume that $f(x)$ is $2π$-periodic and square integrable on $(−π, π)$.
Can someone help me show how following statements are true using Bessel's Inequality?
$$\lim_{n\rightarrow \infty}\int_{-\pi}^{\pi}f(x)\cos(nx)dx=0$$ $$\lim_{n\rightarrow \infty}\int_{-\pi}^{\pi}f(x)\sin(nx)dx=0~.$$
For each $n\in \mathbb{N}$, let $e_n(x)=\cos(nx)$. Then $\{e_n : n\in \mathbb{N}\}$ is an orthonormal family of functions in $L^2[-\pi, \pi]$. Hence, for any $f\in L^2[-\pi, \pi]$: $\sum\limits_{n=1}^\infty |\langle f, e_n\rangle|^2 \leq ||f||_2$. In particular we must hvae $|\langle f, e_n\rangle| \rightarrow 0$ which is exactly what you want. The proof is the same for the $\sin$ functions.