Subset dense in R

  • #1

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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Answers and Replies

  • #2
tiny-tim
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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! :smile:

With questions like this, always start with the definition

what definition has your professor given you for a dense subset?​
 
  • #3
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0

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.
 
  • #4


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.
 
  • #5
tiny-tim
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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:
 
  • #6
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?
 
  • #7
tiny-tim
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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:
 
  • #8
tiny-tim
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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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