This is from Fitzpatrick's Advanced Calculus, where it has already been shown that the rationals are dense in $\displaystyle \mathbb{R}$:

Prove that the set $\displaystyle \mathbb{Q}\backslash\mathbb{Z}$ of rational numbers that are not integers is dense in $\displaystyle \mathbb{R}$.So far I have:

We know the rationals $\displaystyle \mathbb{Q}$ are dense in $\displaystyle \mathbb{R}$. In the case where there are no integers in the set $\displaystyle (a,b)$ where $\displaystyle a<b$, then $\displaystyle (a,b)\backslash\mathbb{N}=(a,b)$ and so $\displaystyle \mathbb{Q}\cap (a,b)\not=0$ as shown previously.

In the case where there is at least one integer $\displaystyle m$ in the set $\displaystyle (a,b)$, we want to show that there is also a non-integer in $\displaystyle (a,b)$.

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!