Convergents of a regular continued fraction

Let $a_0,a_1,\dots$ be the digit sequence of a regular continued fraction (Finite and infinite regular continued fractions), terminating at $a_N$ in the finite case. Define integers $p_n,q_n$ by $p_{-2}=0,p_{-1}=1,q_{-2}=1,q_{-1}=0,$ and, for every index $n\ge 0$ for which the digit $a_n$ exists, $p_n=a_n{p}_{n-1}+p_{n-2},q_n=a_n{q}_{n-1}+q_{n-2}\mathrm{.}$

The denominators are positive at every digit index: $q_0=1$, and if the digit $a_1$ exists then $q_1=a_1>0$; thereafter $q_n=a_n{q}_{n-1}+q_{n-2}>0$ by induction because $a_n>0$ and the preceding denominators are nonnegative. Thus the quotient below is defined in $\mathbb{Q}$.

The rational number $p_n\mathrm{/}{q}_n$ is the $n$-th convergent. The initial labels $-2$ and $-1$ belong only to this recurrence convention; they do not extend the digit sequence itself to negative indices.