A Elementary fact but proof needed

Question

Let $n,q\in\mathbb{N}$, $r\in\mathbb{R}$ and $m,p\in\mathbb{Z}$ such that $\frac{m}{n}<r<\frac{m+1}{n}$ and $|\frac{p}{q}-r|<\min(r-\frac{m}{n};\frac{m+1}{n}-r)$. It does seem obvious that we should have $q>n$, but am having trouble showing this trivial observation, so and help and hints will be greatly appreciated. I tried showing this via contradiction, but seem to get no where. Thanks.

Answer 1

$$\frac14<\frac13<\frac24$$