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

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Homework Help Overview

The discussion revolves around proving that the set of rational numbers of the form P/2^n, where n and p belong to the integers, is dense in the real numbers. Participants are exploring the concept of density in the context of rational numbers and real analysis.

Discussion Character

  • Conceptual clarification, Assumption checking, Exploratory

Approaches and Questions Raised

  • Participants are questioning how to demonstrate that a set is dense in R, with references to definitions and the need for a formal approach. There are attempts to clarify the definition of a dense subset and how to apply it to the given set of rational numbers.

Discussion Status

The discussion is ongoing, with participants providing hints and prompting each other to think about definitions and the standard ε-δ proof method. There is a focus on ensuring understanding of the concepts involved, but no consensus or resolution has been reached yet.

Contextual Notes

Some participants express uncertainty about the definitions and the implications of the parameters p and n being integers. There is also a mention of needing to choose an appropriate ε for the proof.

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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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