Proof of Fejer's lemma

Question

How does ont prove Fejer's lemma:

If $f \in L^1(\mathbb{T})$ and $g\in L^\infty(\mathbb{T})$, then $\lim_{n \rightarrow \infty} \int f(t) g(nt) \, dt = \hat{f}(0)\hat{g}(0).$

Answer 1

Show that it's enough to prove it assuming $f$ is a trigonomtric polynomial. Figure out the Fourier coefficients of $g(nt)$ in terms of $\hat g(k)$.