A consequence of Riemann-Lebesgue lemma Riemann Lebesgue lemma states that: For a function $f\in L_1([-\pi,\pi])$ we have 
\begin{align}
\lim_{|n|\to \infty}  \int_{[-\pi,\pi]} f(t) e^{-int} \text{ d}t =0.
\end{align}
The integral here is Lebesuge integral. As consequences of this lemma, 
\begin{align}
\lim_{|n|\to \infty} \int_{[-\pi,\pi]} f(t) \cos(nt) \text{ d}t =0 ,
\end{align}
and
\begin{align}
\lim_{|n|\to \infty} \int_{[-\pi,\pi]} f(t) \sin(nt) \text{ d}t = 0.
\end{align}
How to prove these two corollaries. I need any help. 
Thanks in advance.

Assuming $\lim_{|n|\rightarrow\infty}\int f e^{inx}dx = 0$ gives
$$
   \lim_{n\rightarrow\infty}\int f e^{inx}dx = 0 \\
   \lim_{n\rightarrow\infty}\int f e^{-inx}dx = 0
$$
Adding these results in
$$
         \lim_{n\rightarrow\infty}\int f \cos(nx)dx = 0
$$
I'll let you tackle the other combinations.
