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Given a continuous function $f$ of period $2\pi$ and $\alpha$ in $\mathbb{R}$

can you show for any real $x$ that $\lim_{ N\to \infty} \sum_{n=1}^N f(x+n\alpha) = \lim_{ N\to\infty} \sum_{n=1}^N f(x_n)$

that is, the 'average value' of $f$ diluted by $N$ is the same as the average value of the sequence $x+n\alpha$ since $\alpha/\pi$ is never irrational

where $\alpha/\pi$ is irrational

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