Posted: Tue Dec 28, 2004 5:50 am    Post subject: A strange feature of function [x]

Hi, could anybody help mi with this problem:

Let be some positive irational numbers that satisfy equation . Prove that for each natural number there exist some natural number such that or . Where denotes the integral part of number , examples: [5,1] = [5,4] = [5,7] = 5.

Thanks._________________

No quiero meter las cabras en el corral a tu.

Posted: Tue Dec 28, 2004 10:42 pm    Post subject:

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.