A limit of integral

Question

How can I prove that

$ \lim_{ n\to \infty} \int_{R}^{} \cos(nt)f(t)dt = 0 $ for any $f \in L_{1}(R) $? I believe that I should use a fact that cosinus is a cyclic function and divide this integral into a series of integrals $ \int_{[\frac{-\pi}{2n} +\frac{2k\pi}{n}] }^{[\frac{-\pi}{2n} +\frac{2k\pi + \pi}{n}]} \cos(nt)(f(t) -f(t+\frac{\pi}{n}))dt$ for $k \in Z$ but I don't know what to do next.Will be grateful for any hint.

Answer 1

For differentiable $f$, do it by parts. Then use that an arbitrary $f\in L^1(\mathbb R)$ is a limit of differentiable functions.