# A question from real analysis

## 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

Related Calculus and Beyond Homework Help News on Phys.org
Dick
Homework Helper
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??

Dick
Homework Helper
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)??

Dick