# Homework Help: Limit of a trigonometric integral

1. Jul 5, 2012

### hofhile

1. The problem statement, all variables and given/known data
Excersice of Big Rudin: ch. 4, prob. 9.

If $$A\subset [0,2\pi]$$ and $$A$$ is measurable, prove that

$$\lim_{n\to\infty}\int_{A}\cos\,nx\,dx=0$$

2. Relevant equations
Bessel's inequality

3. The attempt at a solution
I give my solution, but I post because I think that my answer is completely wrong:

$$\chi_{A}\in L^{2}(T)$$ so $$c_{n}=\frac{1}{2\pi}\int_{A}e^{-int}\,dt$$ are the Fourier coefficient and by Bessel's inequality $$\sum_{n=-\infty}^{\infty}|c_{n}|^{2}<\infty$$ therefore $$|c_{n}|\to 0$$ and $$\lim_{n\to\infty}\int_{A}\cos\,nx\,dx=0$$.

Thank you.

Last edited: Jul 5, 2012