measure theory - Prove that $ lim_{n to infty} int_{-infty}^{infty} sin(nt) f(t) d t = 0 $.

I am trying to prove that $ \lim_{n \to \infty} \int_{-\infty}^{\infty} \sin(nt) f(t) d t = 0 $ for every Lebesgue integrable function $ f $ on $ \mathbb{R} $. My first thoughts were to use Dominated Convergence Theorem but I realised that there is no pointwise limit of the sequence of functions $ f_n = \sin(nt) f(t) $. I do not know how to proceed.

Any help would be appreciated. Thanks!