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.