Define the continued fraction expansion of $\theta \in \mathbb{R}$, and show that this expansion terminates if and only if $\theta \in \mathbb{Q}$.

Define the convergents $\left(p_{n} / q_{n}\right)_{n \geqslant-1}$ of the continued fraction expansion of $\theta$, and show that for all $n \geqslant 0$,

$$p_{n} q_{n-1}-p_{n-1} q_{n}=(-1)^{n-1} .$$

Deduce that if $\theta \in \mathbb{R} \backslash \mathbb{Q}$, then for all $n \geqslant 0$, at least one of

$$\left|\theta-\frac{p_{n}}{q_{n}}\right|<\frac{1}{2 q_{n}^{2}} \quad \text { and } \quad\left|\theta-\frac{p_{n+1}}{q_{n+1}}\right|<\frac{1}{2 q_{n+1}^{2}}$$

must hold.

[You may assume that $\theta$ lies strictly between $p_{n} / q_{n}$ and $p_{n+1} / q_{n+1}$ for all $n \geqslant 0 .$ ]