Let $\theta \in \mathbb{R}$.

For each integer $n \geqslant-1$, define the convergents $p_{n} / q_{n}$ of the continued fraction expansion of $\theta$. Show that for all $n \geqslant 0, p_{n} q_{n-1}-p_{n-1} q_{n}=(-1)^{n-1}$. Deduce that if $q \in \mathbb{N}$ and $p \in \mathbb{Z}$ satisfy

$$\left|\theta-\frac{p}{q}\right|<\left|\theta-\frac{p_{n}}{q_{n}}\right|$$

then $q>q_{n}$.

Compute the continued fraction expansion of $\sqrt{12}$. Hence or otherwise find a solution in positive integers $x$ and $y$ to the equation $x^{2}-12 y^{2}=1$.