I would like to prove that given any $\alpha = [a_0; a_1 \ldots] \in \mathbb{R}$, with: \begin{equation} \alpha_n = [a_n; a_{n-1}, \ldots, a_0]\\ \frac{p_n}{q_n} = [a_0; a_1, \ldots, a_n] \end{equation} Then we have the following; \begin{equation} q_n \alpha - p_n = \frac{(-1)^n}{\alpha_{n+1}q_n +q_{n-1}} \end{equation}
I have started by deriving the relationship that $\alpha_n = \frac{q_n}{q_{n-1}}$, but I'm not sure if (or how) this might help. Also, I know that \begin{equation} p_n q_{n-1} - p_{n-1}q_n = (-1)^{n+1} \end{equation} Any suggestions are appreciated.