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!

  • $\begingroup$ Did you look at trigonometric polynomials? That is, did you explicitly compute this quantity for a trigonometric polynomial? $\endgroup$ – user296602 Dec 11 '15 at 5:45
  • $\begingroup$ Oh, I forgot to add what I've done. Let me update it now. $\endgroup$ – Gael Diego Fernandez Dec 11 '15 at 5:46

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