Let $p_N/q_N$ be the $N^\text{th}$ convergent of the continued fraction for some irrational number $\alpha$. It turns out that for any other approximation $p/q$ (with $q \le q_N$) which isn't a convergent $|\alpha q - p| > |\alpha q_{N-1} - p_{N-1}|$. I'm wondering if there are any nice proofs for this result?
In my book this is proved by picking $x,y$ that solves
$$ \begin{pmatrix} p_N & p_{N-1} \\ q_N & q_{N-1} \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} p \\ q \end{pmatrix} $$
since $x$ and $y$ have opposite sign, as well as $\alpha q_N - p_N$ and $\alpha q_{N-1} - p_{N-1}$ have opposite sign we can conclude that $|\alpha q - p| = |x (\alpha q_N - p_N) + y (\alpha q_{N-1} - p_{N-1})| = |x| |\alpha q_N - p_N| + |y| |\alpha q_{N-1} - p_{N-1}|$ which proves the theorem.
I am looking for different proofs than this one.