$ \sum a_n \sin (nx)$ converges uniformly iff $na_n\to 0$ as $n \to \infty$ 
Let $\{a_n\}$ be decreasing sequence of positive terms then prove that $\displaystyle \sum a_n \sin (nx)$ converges uniformly on $\Bbb{R}$ iff $na_n \to 0$ as $n\to \infty$.

I proved that convergence of $ \sum a_n \sin (nx)$ implies $na_n \to 0$ as $n\to \infty$ . I got stuck while prove the converse,  I tried ussing Dirichlet's test but the problem here is that partial sums of  $\displaystyle \sum^{n}_{k=1} \sin (kx)$ are bounded by $\displaystyle \frac{1}{|\sin (\frac{t}{2})|}$ , so as per Dirichlet's test requirement I'm not getting uniform bound.

If:
$$ D_N(x) = \sum_{n=1}^{N}\sin(nx) $$
then we have:
$$\left| D_N(x) \right|\leq \min\left(N,\frac{1}{|\sin(x/2)|}\right) $$
and the claim follows by partial summation.
