This is from Fitzpatrick's Advanced Calculus, where it has already been shown that the rationals are dense in :

Prove that the set of rational numbers that are not integers is dense in .So far I have:

We know the rationals are dense in . In the case where there are no integers in the set where , then and so as shown previously.

In the case where there is at least one integer in the set , we want to show that there is also a non-integer in .

I am not sure where to go from here, or if dividing it into cases is the correct approach. Could anyone give me some pointers? Thanks!