Convergence of a positive simple continued fraction
= Convergence of a positive simple continued fraction
When $b_j=1$ and $a_j$ are positive integers, adjacent convergents satisfy
$$
p_nq_{n-1}-q_np_{n-1}=(-1)^{n+1}.
$$
Even and odd convergents are monotone from opposite sides, their separation is $1/(q_nq_{n+1})$, and $q_n\to\infty$. They therefore have a common limit $x$, with
$$
\left|\frac{p_n}{q_n}-x\right|
+\left|\frac{p_{n+1}}{q_{n+1}}-x\right|
=\frac1{q_nq_{n+1}}.
$$