TITLE: Show that $\lim_{n\to\infty}\frac{\lfloor nx \rfloor}{n}=x$
QUESTION [6 upvotes]: How do I prove that $\lim_{n\to\infty}\frac{\lfloor nx \rfloor}{n}=x$ for $x\in\mathbb{R}$? I see that $\lfloor nx\rfloor = n\lfloor x \rfloor + \lfloor n(x-\lfloor x \rfloor )\rfloor + O(1)$ but I'm not sure how to deal with the middle term.

REPLY [5 votes]: Why not start from $nx=\lfloor nx\rfloor +e_n(x)$, where $0\le e_n(x)\lt 1$?