# Subset dense in R

## Homework Statement

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?

## Homework Equations

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

## The Attempt at a Solution

I do not know how to check dense in R!

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tiny-tim
Homework Helper
Welcome to PF!

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!

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

## Homework Statement

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?

## Homework Equations

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

## The Attempt at a Solution

I do not know how to check dense in R!

You need to think about how R is defined.

Hi Hamed! Welcome to PF!

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
Homework Helper
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

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

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

tiny-tim
Homework Helper
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.

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

tiny-tim
Homework Helper
Hi Hamed! Thanks for the PM.

(copy my "ε"! )
HamedJafarian said:
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 … ?