Beatty's Theorem

Theorem
Let $r, s \in \R \setminus \Q$ be an irrational number such that $r > 1$ and $s > 1$.

Let $\mathcal B_r$ and $\mathcal B_s$ be the Beatty sequences on $r$ and $s$ respectively.

Then $\mathcal B_r$ and $\mathcal B_s$ are complementary Beatty sequences :
 * $\dfrac 1 r + \dfrac 1 s = 1$

Also known as
Beatty's theorem is also known as the Rayleigh theorem, for.