Certain Fraction between Fractions

Question

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 $r,s\in\mathbb{Z}$?

Answer 1

Since $0\lt p\lt q-1$, we have $$ \frac pq\lt\frac p{q-1}\lt\frac{p+1}q $$