A question from real analysis

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Homework Statement


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.


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  • #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.
 
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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??
 
  • #4
Dick
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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??
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.
 
  • #5
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hmmm, like epsilon/(n^2+n)??
 
  • #6
Dick
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hmmm, like epsilon/(n^2+n)??
Sure, that works. I would have said sum epsilon*(1/2)^n. But whatever you like.
 

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