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

(b) Consider the continued fraction expression

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

in which the coefficients $a_{n}$ are all positive integers forming an unbounded set. Let $\frac{p_{n}}{q_{n}}$ be the $n$th convergent. Prove that

$$\left|x-\frac{p_{n}}{q_{n}}\right| \leqslant \frac{1}{q_{n} q_{n+1}}$$

and use this inequality together with Liouville's theorem to deduce that $x^{2}$ is irrational.

[ You may assume without proof that, for $n=1,2,3, \ldots$,

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