TITLE: Certain Fraction between Fractions
QUESTION [6 upvotes]: Is there always a fraction $\frac{r}{s}$ with $\frac{p}{q}<\frac{r}{s}<\frac{p+1}{q}$ and $s<q$ for $0<p<q-1\in\mathbb{Z}$ and $r,s\in\mathbb{Z}$?

REPLY [2 votes]: Since $0\lt p\lt q-1$, we have
$$
\frac pq\lt\frac p{q-1}\lt\frac{p+1}q
$$