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Homework Help: A question from real analysis

  1. Mar 23, 2010 #1
    1. The problem statement, all variables and given/known data
    Prove that the rationals as a subset of the reals can all be contained in open intervals the sum of whose width is less than any \epsilon > 0.


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 23, 2010 #2

    Dick

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    You aren't playing the game here. What do you think about the problem? You can't leave the Attempt at a Solution completely blank.
     
  4. Mar 23, 2010 #3
    ups, my bad. I was thinking that since rationals are countable, I just need to make the open interval around each rational such that the total sum is less than \epsilon. Then the question turns to prove the sum of a finite series converges to the epsilon??
     
  5. Mar 23, 2010 #4

    Dick

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    Much better, thanks. Can you write a series that converges to epsilon? If you can write a series that sums to say, 1, you should be able to write a series that converges to epsilon.
     
  6. Mar 23, 2010 #5
    hmmm, like epsilon/(n^2+n)??
     
  7. Mar 23, 2010 #6

    Dick

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    Sure, that works. I would have said sum epsilon*(1/2)^n. But whatever you like.
     
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