The problem:
Let $h$ and $k$ be positive integers. Prove that for every $\varepsilon>0$, there are positive integers $m$ and $n$ such that $$ \varepsilon<|h \sqrt{m}-k \sqrt{n}|<2 \varepsilon $$
My approach: note it suffices to find $\frac{m}{n} \approx \frac{k^2}{h^2}$. Because the latter is a real number, we can use the density of the rationals to approximate it arbitrarily well, and so in particular we can find an $m, n \in \mathbb{N}$ that is a $2\epsilon$-approximation, but not an $\epsilon$-approximation. I am trying to formalize this intuition. I know the triangle and reverse triangle inequalities will probably be needed. I'm starting from the fact that I know that I have an $m, n$ such that $\frac{m}{n} \in [\frac{k^2}{h^2}, \frac{k^2}{h^2} + \epsilon]$, and trying to conclude, but am unable.