Can you solve Problem 19 from Chapter 8 of Rudin's Principles of Mathematical Analysis, I'm having a lot of difficulty with it
I've proven the first part, namely $$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N \exp(ik(x+n\alpha))=\frac{1}{2\pi}\int_{-\pi}^\pi(\cdots) = \begin{cases} 1\text{ if }k=0\\0\text{ otherwise}\end{cases}$$
Now I want to prove that if $f$ is continuous in $\mathbb{R}$ and $f(x+2\pi)=f(x)$ for all $x$ then
$$\lim_{N\to\infty} \sum_{n=1}^{N} \frac{1}{N} f(x+n\alpha)=\frac{1}{2\pi} \int\limits_{-\pi}^{\pi}f(t)\mathrm dt$$
for any $x$, where $\alpha/\pi$ is irrational.
I've tried writing it as
$$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N \sum_{k=0}^N\frac{1}{2\pi}\int_{-\pi}^\pi e^{ikt}f(x+n\alpha) $$ but that was not helpful.