Let $(a_k)_{k\geq 0}$ be a continued-fraction partial-quotient sequence with $a_0\in\mathbb{Z}$ and $a_k\in\mathbb{N}$ for every $k\geq 1$. Let $(p_k)_{k\geq -1}$ and $(q_k)_{k\geq -1}$ be the convergent numerator and denominator sequences defined by
at $m=a_{n+1}$. For $0<m<a_{n+1}$ they lie strictly between these endpoints, and the whole list moves monotonically as $m$ increases. Their denominators are $mq_n+q_{n-1}$.
paragraph
admin
For $n=0$, the same monotonicity statement holds for the admissible values $1\le m\le a_1$, with positive denominators $m$ and endpoint