About Uniform Convergence of $\sum_{n=1}^\infty\frac{\sin nx}{n}$ on $[0,2\pi]$

Question

Is $\sum_{n=1}^\infty\dfrac{\sin nx}{n}$ uniform convergent on $[0,2\pi]$?

I think it is not. However, I could not prove it by Cauchy's criterion.

Answer 1

A related problem.

Hint: Notice that the series is the Fourier series of the function

$$ -\frac{\pi+x}{2}. $$

Now, use the following result:

Theorem: The Fourier series of a $2\pi$-periodic continuous and piecewise smooth function converges uniformly.