This is from I. M. Gelfand's Algebra book.
Fractions $\displaystyle\frac{a}{b}$ and $\displaystyle\frac{c}{d}$ are called neighbor fractions if their difference $\displaystyle\frac{ad - bc}{bd}$ has numerator of $\pm 1$, that is, $ad - bc = \pm 1$. Prove that
(a) in this case neither fraction can be simplified (that is, neither has any common factors in numerator and denominator);
(b) if $\displaystyle\frac{a}{b}$ and $\displaystyle\frac{c}{d}$ are neighbor fractions, then $\displaystyle\frac{a+c}{b+d}$ is between them and is a neighbor fraction for both $\displaystyle\frac{a}{b}$ and $\displaystyle\frac{c}{d}$; moreover,
(c) no fraction $\displaystyle\frac{e}{f}$ with positive integer $e$ and $f$ such that $f < b + d$ is between $\displaystyle\frac{a}{b}$ and $\displaystyle\frac{c}{d}$.
Parts (a) and (b) weren't too difficult, but I'm stuck on part (c). I've included (a) and (b) in case they're related to the solution to (c).