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Subset dense in R

  1. Sep 7, 2009 #1
    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!
     
    Last edited by a moderator: Sep 7, 2009
  2. jcsd
  3. Sep 7, 2009 #2

    tiny-tim

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    Welcome to PF!

    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?​
     
  4. Sep 7, 2009 #3
    You need to think about how R is defined.
     
  5. Sep 8, 2009 #4
    Re: Welcome to PF!

    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.
     
  6. Sep 8, 2009 #5

    tiny-tim

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    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:
     
  7. Sep 8, 2009 #6
    Is it correct for when the p ,n are blong to Z?
    and with is it m?
     
  8. Sep 8, 2009 #7

    tiny-tim

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    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:
     
  9. Sep 8, 2009 #8

    tiny-tim

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    Hi Hamed! Thanks for the PM. :smile:

    (copy my "ε"! :wink:)
    Choose m so that 1/2m < ε,

    and then … ? :smile:
     
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