The result is essentially a Corollary of the Beatty’s Theorem. The Beatty’s Theorem states that:
Given irrational numbers , satisfy . Then and .
The proof of the theorem is elementary. For every natural number , let and be the greatest integers satisfying and . Then the number of elements in and that is less than or equal to is equal to: . By the inequalities: , , , yield .
Using Induction on N and the result follows. |