Prove that for $\alpha\not\in\mathbb{Q},\alpha>0$, $([n+n\alpha])_{n\in\mathbb{N}}\sqcup([n+n\alpha^{-1}])_{n\in\mathbb{N}}=\mathbb{N} $ My Quesion: Prove that for $\alpha\not\in\mathbb{Q},\alpha>0$, $([n+n\alpha])_{n\in\mathbb{N}}\bigcup([n+n\alpha^{-1}])_{n\in\mathbb{N}}=\mathbb{N}$, where $[k]$ means the integral part of $k\in\mathbb{Z}$, and $\sqcup$ means disjoint union.
My Idea: I just realized that $$\frac{1}{1+\alpha}+\frac{1}{1+\alpha^{-1}}=1, \quad\text{where } 1+\alpha, 1+\alpha^{-1} \text{ are positive irrational numbers}$$
So the interested sequences are Beatty's sequences. And here is a  proof (page 11-13) for Beatty sequences.

"You already provided a complete solution to your problem" Yes, I just proved it.
