If $\alpha$ is an irrational number and $\frac{p_n}{q_n}$ for $n\geqslant 0$ are the convergents to $\alpha$.
How can one show that $p_{n+1}q_n - p_nq_{n+1} = \pm 1$ for all $n \geqslant 0$, where:
I have attempted to show this by induction assuming it holds for every pair up to some pair $n=k$ but there doesn’t appear to be a clear way to simplify the blue terms at the end of the inductive step.
