Fejér is Theorem (Problem in Rudin)

Can you address Problem 19 from Chapter 8 of Rudin is Principles of Mathematical Analysis, I'm having a great deal of trouble with it

I've confirmed the first component, particularly $$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N \exp(ik(x+n\alpha))=\frac{1}{2\pi}\int_{-\pi}^\pi(\cdots) = \begin{cases} 1\text{ if }k=0\\0\text{ otherwise}\end{cases}$$

Now I intend to confirm that if $f$ is continual in $\mathbb{R}$ and also $f(x+2\pi)=f(x)$ for all $x$ after that

$$\lim_{N\to\infty} \sum_{n=1}^{N} \frac{1}{N} f(x+n\alpha)=\frac{1}{2\pi} \int\limits_{-\pi}^{\pi}f(t)\mathrm dt$$

for any kind of $x$, where $\alpha/\pi$ is illogical.

I've attempted creating it as

$$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N \sum_{k=0}^N\frac{1}{2\pi}\int_{-\pi}^\pi e^{ikt}f(x+n\alpha) $$. yet that was not handy.

2019-05-19 00:09:55
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