Proving Density in R: Subset of Rational Numbers P/2^n for n,p belong to Z

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SUMMARY

The discussion focuses on proving that the set of rational numbers of the form P/2^n, where n and p belong to the integers (Z), is dense in the real numbers (R). Participants emphasize starting with the definition of a dense subset, which states that for every x in R, there exists a y in the subset that is arbitrarily close to x. A key approach involves using the epsilon-delta definition of limits, specifically choosing epsilon as 1/2^m to demonstrate the density of the set.

PREREQUISITES
  • Understanding of dense subsets in real analysis
  • Familiarity with rational numbers and their properties
  • Knowledge of the epsilon-delta definition of limits
  • Basic concepts of set theory and integers (Z)
NEXT STEPS
  • Study the definition of dense subsets in real analysis
  • Learn about the epsilon-delta proof technique in calculus
  • Explore the properties of rational numbers and their density in R
  • Investigate examples of dense subsets in various mathematical contexts
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Mathematics students, particularly those studying real analysis, educators teaching concepts of density in sets, and anyone interested in the properties of rational numbers within the real number system.

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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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Welcome to PF!

HamedJafarian said:
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?​
 
HamedJafarian said:

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.
 


tiny-tim said:
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.
 
HamedJafarian said:
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:
 
tiny-tim said:
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?
 
HamedJafarian said:
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:
 
Hi Hamed! Thanks for the PM. :smile:

(copy my "ε"! :wink:)
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 … ? :smile:​
 

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