We know that \begin{equation*} a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cfrac{1}{\ddots+\cfrac{1}{a_n}}}}}=[a_0,a_1, \cdots, a_n] \end{equation*}
If $\frac{p_n}{q_n}=[a_0,a_1, \cdots, a_n]$.
Let us define $$\lim_{n \to \infty}\frac{p_n}{q_n}=x$$
How to prove that $$x=\frac{p_n+p_{n-1}x_{n+1}}{p_n+p_{n-1}x_{n+1}}$$ where $x_n=[a_n,a_{n+1},\cdots]$
Yes, there is a misprint here 2nd page last equation. Can anyone rectify it and prove it.