TITLE: Show that $\int_{-\pi}^\pi ~f(x) \cos (nx) \mathrm{d}\mu(x)$ converges to $0$
QUESTION [7 upvotes]: I need some help on the following problem.

Let $f\in L_1([-\pi,\pi])$. Then $\int_{-\pi}^\pi ~f(x) \cos (nx) \mathrm{d}\mu(x) \to 0$, where $\mu$ is the Lebesgue measure on $[\pi,\pi]$. 

Any hints and suggestions on how to begin is very much welcomed. Thanks.

REPLY [6 votes]: This statement is true for any $f\in C^1([-\pi,\pi])$. Note that $C^1([-\pi,\pi])$ is dense in $L_1([-\pi,\pi])$ hence equality holds for all $f\in L_1([-\pi,\pi])$