I feel my question is a symptom of a larger issue I have with analysis, which is being unable to get the exact proof even when you have the right idea. It's also an issue of knowing how much rigour is necessary.
Here is my proof so far (with the right idea of taking the difference of the numbers and multiplying by a large enough integer so there is an integer between them):
suppose $ 0 < x < y, \ x, y\in \mathbb{R} $ (proof can be extended to negative reals easily)
there exists $ N \in \mathbb{N} $ s.t. $ N(y-x) > 1 \Leftrightarrow Ny - 1 > Nx $
$ Ny \geq \lfloor Ny \rfloor \geq Ny - 1 > Nx, \lfloor Ny \rfloor \in \mathbb{Z} $
$ x < \frac{\lfloor Ny \rfloor}{N} \leq y $
...and the issue here is the inequality is not strict. I know for a fact if the difference of two numbers is greater than 1, then there is an integer between them, but saying 'take m in Z between Nx and Ny' to me seems unrigorous. Maybe you should prove it? Maybe it is fine to say? This is the issue I have with analysis, I'm not sure of the amount of rigour necessary, especially important in an exam. But maybe in the same vein I should be questioning if we have to prove we can multiply the difference to get it larger than one (but I think that's fine). Let me know if it was a silly idea to use the floor function.