Proving the uniform convergence of the average sequence of $f_n(x)=\sin(nx)$

Question

I was asked to prove that:

1) $f_n(x)=\sin(nx)$ does not converge pointwise.

2) The average sequence of $f_n(x)=\sin(nx)$ is uniformly convergent.

I secceed to prove the first part but I cannot prove the other one. In addition it is not allowed to use the M-test.

Thanks. (I do not know how to use the function signs.)

Answer 1

Hint:

$$\frac1n\sum_{k=0}^{n-1} e^{ikx}=\frac{e^{inx}-1}{n(e^{ix}-1)}.$$