Subset dense in R

[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!

[tiny-tim]

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?

Hi Hamed! Welcome to PF! :smile:

With questions like this, always start with the definition …

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

[wofsy]

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!

You need to think about how R is defined.

[HamedJafarian]

Hi Hamed! Welcome to PF! :smile:

With questions like this, always start with the definition …

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

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.

[tiny-tim]

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.

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 :wink:

[HamedJafarian]

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 :wink:

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

[tiny-tim]

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

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

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

[tiny-tim]

Hi Hamed! Thanks for the PM. :smile:

(copy my "ε"! :wink:)
I mean that i must show that for every eps and x, there is a y that y-x<eps.how can i show this one?


Choose m so that 1/2m < ε,

and then … ? :smile: