Let $f$ be a $\mathcal{C}^r$ function such that $f(0)=f(\pi)=0$, and define $a_n := \frac{2}{\pi}\int^\pi_0 sin(nx)f(x)dx$, its easy to show that exists $C>0$ such that $|a_n|\leq \frac{C}{n^r}$. Its possible to find $K>0$ such that $|a_n| \leq \frac{K}{n^{r+1}}$? In order to do this i only have too bound $\frac{2}{\pi}\int^\pi_0 sin(nx)f(x)dx$ by some $\frac{K}{n}$ for $f$ continuous. How can i prove this last fact?
Thanks in advance.