A Density of Irrational Numbers in [0;1]

[Calabi]

Hello, let be $x \in \mathbb{R} - \mathbb{Q}$, do we have de density of $\{nx - \lfloor{nx} / n \in \mathbb{N}\}$ in $[0; 1]$ please?

I think yes but it's just an intuition : if I take a and b in $[0; 1]$ with a < b, I have an irrationnal between them let call it c but I don't know hat to do with it.

Could you help me please?

Thank you in advance and have a nice afternoon.

 

[fresh_42]

In principle you already have said what you need. How is a dense subset formally defined? It might be a bit of work to formally prove that there are always irrational numbers as close to a given rational one as wanted. But given this fact, it's basically the definition of density itself.
E.g.: https://en.wikipedia.org/wiki/Dense_set

 

[Calabi]

No that's not what I mean : the irrationnal number are dense in $\mathbb{R}$ so in $[0; 1]$ too (1)
And I wanted to use that to show that
$\{nx - \lfloor nx \rfloor / n \in \mathbb{N}\}$ in [0; 1], where x is an irrationnal number, by using (1).

 

[micromass]

http://press.princeton.edu/chapters/s12_8220.pdf Theorem 12.2.2

 

[fresh_42]

http://press.princeton.edu/chapters/s12_8220.pdf Theorem 12.2.2

That's what I was looking for. Thank you.