(i) Let $\alpha$ be an algebraic number and let $p$ and $q$ be integers with $q \neq 0$. What does Liouville's theorem say about $\alpha$ and $p / q$ ?

(ii) Let $p$ and $q$ be integers with $q \neq 0$. Prove that

$$\left|\sqrt{2}-\frac{p}{q}\right| \geqslant \frac{1}{4 q^{2}} .$$

[In (ii), you may not use Liouville's theorem unless you prove it.]