Prove that the Fourier transform of an integrable function $f$ satisfies:
$$\lim_{|n| \rightarrow \infty} c_{n}(f) = 0$$
where the Fourier transform $\hat{f}$ of an integrable function $f$ is defined as the sequence $(c_{n}(f))_{n \in \mathbb{Z}}$, where $c_{n}(f)$ is the Fourier coefficient of a function $f \in L^{1}([-L,L])$, and is defined to be $$c_{n}(f) = \frac{1}{2L}\int^{L}_{-L} f(x) \overline{g_{n}(x)}dx = \langle f,g_{n}\rangle.$$ And for $L>0$ and for $n \in \mathbb{Z}$ we take $$g_{n}(x) = e^{in\pi x/L}. $$
Could anyone give me a hint, please?
This is the Riemann-Lebesgue Lemma. It's in lots of books. One way to prove it is to approximate $f$ by a step function, and for step functions it can be proved by direct calculation.