Let $\frac{a}{b}$ and $\frac{c}{d}$ be two reduced fractions with $bc-ad > 1$ (and hence $\frac{a}{b} \lt \frac{c}{d}$) and $a,b,c,d$ positive. It is well known that there are integers $u,v$ such that $0 \leq u \leq a, 0 \leq v \leq b$ and $av-bu=1$. Let $q$ be the largest positive below $\frac{ \max(b,d)+v}{b}$, i.e. the euclidean quotient of $\max(b,d)+v$ by $b$). Then the following inequality holds :
$ cv-du \leq q(bc-ad) \tag{*} $
I know that (*) is true, but the only proof I know relies on the heavy machinery of Farey sequences and the Stern-Brocot tree. Is there a more direct proof ?