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Sagar Kolte
science forum beginner

Joined: 14 Apr 2005
Posts: 31

Posted: Thu Jun 22, 2006 4:27 pm    Post subject: Density of a certain set.

let 'a' be an irrational number.

consider the set {n*a|n is a natrual number}

prove that the set {x|x= n*a - [n*a]} is dense in [0 1].

note: [y]=gratest integer less than or equal to y.

Thank You.
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*a|n is a natrual number} prove that the set {x|x= 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...

 Quote: Thank You.

************************

David C. Ullrich

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