# A question from real analysis

• zhang128
In summary, the problem is to prove that the rationals can be contained in open intervals with a total width less than any epsilon. To do this, one can create a series that converges to epsilon, such as epsilon/(n^2+n) or epsilon*(1/2)^n.

## 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.

## The Attempt at a Solution

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.

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??

zhang128 said:
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.

hmmm, like epsilon/(n^2+n)??

zhang128 said:
hmmm, like epsilon/(n^2+n)??

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