Let $f$ be integrable function. I wanted to prove that $\lim_{s \to \infty}\int_0^{2\pi}f(t)\sin(st)dt=0$.
Firstly I wanted to do it for every simple function. I started with intervals. When $f$ is a characteristic function of interval, the result follows easily.
Then my textbook says that from this result follow for every simple functions and I can't see it.
I wanted to prove it by pi-lambda theorem by considering set of intervals as $\pi$ system and {$A \subset [0,2\pi] : \lim_{s \to \infty}\int_Af(t)\sin(st)dt=0$} as lambda system, however I can't prove it is lambda system. More specifically I can't prove third axiom of lambda system, the one with the union of increasing family of sets.
It would be true if $\lim_{s \to \infty} \sum_{i=1}^\infty \int_{A_{i+1}-A_i}\sin(st)dt=0$ and this would be true if I could change order of sum and limit but I can't see why I can.
Is this result even true? Should I try it in different way?