Sum of a periodic function

Dear **MHF** members,

** Question**. Suppose that $\displaystyle f:\mathbb{R}\to\mathbb{R}$ is a continuous periodic function with a period of $\displaystyle 2\pi$. If $\displaystyle \alpha/\pi$ is irrational, then for any $\displaystyle x\in\mathbb{R}$ show that the sum $\displaystyle \frac{1}{N}\sum_{n=1}^{N}f(x+n\alpha)$ converges to $\displaystyle \frac{1}{2\pi}\int_{-\pi}^{\pi}f(u)\mathrm{d}u$.

Thanks.

**bkarpuz**

Exercise 19 in Chapter 3 of W. Rudin, *Principles of Mathematical Analysis*, McGraw-Hill Science/Engineering/Math, 3rd edition, 1976.