I am studying for my final and am stumped on this problem. Can someone give me hints or post a detailed solution?
Suppose that $f$ is a continuous function on $\mathbb{R}$, with period $1$. Prove that $$\lim_{N\to \infty}\frac{1}{N}\sum_{n=1}^Nf(nx)=\int_0^1f(t)dt$$ for every irrational number $x\in\mathbb{R}$.
The hint says to first look at trigonometric polynomials. Thanks!