# Subset dense in R

1. Sep 7, 2009

### HamedJafarian

1. The problem statement, all variables and given/known data
How can I prove that the set of rational rational of the form P/2^n for n,p belong to Z is dense in R?

2. Relevant equations

How can I prove that a set is dense in R?

3. The attempt at a solution
I do not know how to check dense in R!

Last edited by a moderator: Sep 7, 2009
2. Sep 7, 2009

### tiny-tim

Welcome to PF!

Hi Hamed! Welcome to PF!

what definition has your professor given you for a dense subset?​

3. Sep 7, 2009

### wofsy

You need to think about how R is defined.

4. Sep 8, 2009

### HamedJafarian

Re: Welcome to PF!

Y is a subset of X,Y is dense in X, if for every x that belog to X, there is y blong to Y that is arbitary close to x.

5. Sep 8, 2009

### tiny-tim

ok … then you need to prove that, for any number x in R, there is a number p/2n arbitrarily close to x.

Hint: choose epsilon = 1/2m

6. Sep 8, 2009

### HamedJafarian

Is it correct for when the p ,n are blong to Z?
and with is it m?

7. Sep 8, 2009

### tiny-tim

sorry, Hamed, I've no idea what you mean.

anyway, I'm talking about the standard δ, ε proof … do you know what that is?

8. Sep 8, 2009

### tiny-tim

Hi Hamed! Thanks for the PM.

(copy my "ε"! )
Choose m so that 1/2m < ε,

and then … ?