David C. Ullrich science forum Guru
Joined: 28 Apr 2005
Posts: 2250

Posted: Fri Jun 23, 2006 10:58 am Post subject:
Re: Density of a certain set.



On Thu, 22 Jun 2006 12:27:37 EDT, koltesagar <sagar.kolte@gmail.com>
wrote:
Quote:  let 'a' be an irrational number.
consider the set {n*an is a natrual number}
prove that the set {xx= n*a  [n*a]} is dense in [0 1].
note: [y]=gratest integer less than or equal to y.

There are interesting ways to do this using Fourier
series... A hint for an elementary solution:
Let x_n = n*a  [n*a].
The fact that a is irrational shows that the x_n are
distinct: x_n <> x_m if n <> m. (<> means "is not equal to").
Let eps > 0. Since the x_n are infinitely many elements of
[0,1) there must exist n,m with n <> m such that
x_n  x_m < eps.
Now...
************************
David C. Ullrich 
