Show that $\lim_{n\to\infty}\frac{\lfloor nx \rfloor}{n}=x$

Question

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.

score 9 · Answer 1 · answered Apr 01 '13 at 21:25

9

Hint Use the squeeze theorem with the inequalites $$nx-1<\lfloor nx\rfloor\leq nx$$

answered Apr 01 '13 at 21:25

score 5 · Answer 2 · answered Apr 01 '13 at 21:26

5

Why not start from $nx=\lfloor nx\rfloor +e_n(x)$, where $0\le e_n(x)\lt 1$?

answered Apr 01 '13 at 21:26

André Nicolas

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Show that $\lim_{n\to\infty}\frac{\lfloor nx \rfloor}{n}=x$

2 Answers2