Taken from Rudin's Real and Complex Analysis text: Suppose $f$ is a continuous function on $\mathbb{R}^1$ with period $1$. Prove that $\displaystyle\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N f(n\alpha) = \int_0^1f(t)dt$ for every irrational real number $\alpha$.
A hint also given is to do it first for $f(t)=\exp(2\pi i k t)$ for $k\in\mathbb{Z}$.
I tried doing this for the function given in the hint, but I end up with $1=0$ in the end (since $\exp(2\pi ik)=1$ for $k\in\mathbb{Z}$, correct?) Anyway, I'm looking for insight as to why I may be getting incorrect answers in my calculations for this hint and also as to how this hint is to help in the solving of the original problem (because I can't seem to make a connection between the two). Thanks in advance!
the integral is zero for the reason you mention. with $\xi=\exp(2\pi ik\alpha)$ the average on the left evaluates to $$ \lim_{N\to\infty}(\xi^N-1)\frac{\xi}{N(\xi-1)} = 0 $$ in fact since for irrational $\alpha$ the values of $f(n\alpha)$ are asymptotically evenly distributed around the circumference of a circle, their mean, in the limit, must be the centre of the circle