(i) State and prove Liouville's theorem on approximation of algebraic numbers by rationals.

(ii) Consider the continued fraction

$$x=\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\frac{1}{a_{4}+\ldots}}}}$$

where the $a_{j}$ are strictly positive integers. You may assume the following algebraic facts about the $n$th convergent $p_{n} / q_{n}$.

$$p_{n} q_{n-1}-p_{n-1} q_{n}=(-1)^{n}, \quad q_{n}=a_{n} q_{n-1}+q_{n-2} .$$

Show that

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

Give explicit values for $a_{n}$ so that $x$ is transcendental and prove that you have done SO.