$S(x)=\sum_{n=1}^{\infty}a_n \sin(nx) $, $a_n$ is monotonic decreasing $a_n\to 0$, when ${n \to \infty}$. I need to prove that for every $\epsilon >0$, the series is uniformly converges within $[\epsilon, 2\pi - \epsilon]$. Can I use Dirichlet and say that $\sum_{0}^{M} \sin (nx)< M$ for every x in the interval and since $a_n$ is uniformly converges to 0 ( uniformity since it does not depend on $x$), so the series is uniform convergent in this range?
In addition I need to prove that if $\sum_{n=1}^{\infty} a_n^2 = \infty$ so the series is not uniform convergent in $[0, 2 \pi]$, Since I know that from $n_0$ and on $a_n^2< a_n$ I used an inequality, again I'm not sure of that.
In the other hand, maybe I need to use Fourier series somehow.
Thanks for the help!