Prove the sequences $\lfloor \alpha n\rfloor $ and $\lfloor \beta n\rfloor $ are disjoint

Question

Here is another problem from a problem set that I can't solve.

Let $\alpha$ and $\beta$ be irrational positive numbers such that $\frac{1}{\alpha}+\frac{1}{\beta}=1$

Prove that the sets $\{ \lfloor \alpha n\rfloor | \;n \in \mathbb N \}$ and $\{ \lfloor \beta n\rfloor | \; n \in \mathbb N \}$ are disjoint and their union is $\mathbb N$.

I tried contradiction to prove they're disjoint, but it's a stalemate.

I don't know why any integer have to be of the form $\lfloor \alpha n\rfloor$ or $\lfloor \beta n\rfloor$ ...

Thanks for any hint

Answer 1

It is known as Beatty theorem or Rayleigh theorem

1). http://www.artofproblemsolving.com/blog/7968

2) http://en.wikipedia.org/wiki/Beatty_sequence

3). https://mathoverflow.net/questions/86516/generalizations-of-the-rayleigh-beatty-theorem

4). http://www.wisdom.weizmann.ac.il/~fraenkel/Papers/betaW6.pdf