Can the Rationals be Contained in Open Intervals with Infinitely Small Width?

[zhang128]

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.

Homework Equations

The Attempt at a Solution

 

[Dick]

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.

 

[zhang128]

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]

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.

 

[zhang128]

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

 

[Dick]

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

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