average of $\{n\alpha \}$, where $\alpha$ irrational

Question

Let $\alpha$ be an irrational number.

Let $\{x\}$ denote the fractional part of $x$.

Is it true that $\displaystyle\lim_{n\to\infty}\frac{\{\alpha \}+\{2\alpha \}+\{3\alpha \}+\cdots+\{n\alpha \}}{n}=\frac{1}{2}$.

How to prove it? Thank you.