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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.

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    @stephen: What is the exercise no.2010-10-22
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    Chandru: It's exercuse #19 in Chapter 82010-10-22
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    @Stephen: Rudin wrote multiple books, what book do you mean?2010-10-22
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    Given what you have proved in the first part, have you tried to apply it to the second part?2010-10-22
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    Hint: Can you prove the relation you are asking about for trigonometric polynomials?2010-10-22
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    @Stephen. Which Rudin's book (and which edition) is this problem from? I looked thru two of my Rudin's books and couldn't find this problem.2010-11-28

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