(a) State Liouville's theorem on the approximation of algebraic numbers by rationals.

(b) Let $\left(a_{n}\right)_{n=0}^{\infty}$ be a sequence of positive integers and let

$$\alpha=a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\ldots}}}$$

be the value of the associated continued fraction.

(i) Prove that the $n$th convergent $p_{n} / q_{n}$ satisfies

$$\left|\alpha-\frac{p_{n}}{q_{n}}\right| \leqslant\left|\alpha-\frac{p}{q}\right|$$

for all the rational numbers $\frac{p}{q}$ such that $0<q \leqslant q_{n}$.

(ii) Show that if the sequence $\left(a_{n}\right)$ is bounded, then one can choose $c>0$ (depending only on $\alpha$ ), so that for every rational number $\frac{a}{b}$,

$$\left|\alpha-\frac{a}{b}\right|>\frac{c}{b^{2}}$$

(iii) Show that if the sequence $\left(a_{n}\right)$ is unbounded, then for each $c>0$ there exist infinitely many rational numbers $\frac{a}{b}$ such that

$$\left|\alpha-\frac{a}{b}\right|<\frac{c}{b^{2}}$$

[You may assume without proof the relation

$$\left.\left(\begin{array}{ll} p_{n+1} & p_{n} \\ q_{n+1} & q_{n} \end{array}\right)=\left(\begin{array}{cc} p_{n} & p_{n-1} \\ q_{n} & q_{n+1} \end{array}\right)\left(\begin{array}{cc} a_{n+1} & 1 \\ 1 & 0 \end{array}\right), \quad n=1,2, \ldots .\right]$$