(i) Let $n \geqslant 4$ be an integer. Show that

$$1+\frac{1}{n+\frac{1}{1+\frac{1}{n+\ldots}}} \geqslant 1+\frac{1}{2 n} .$$

(ii) Let us say that an irrational number $\alpha$ is badly approximable if there is some constant $c>0$ such that

$$\left|\alpha-\frac{p}{q}\right| \geqslant \frac{c}{q^{2}}$$

for all $q \geqslant 1$ and for all integers $p$. Show that if the integers $a_{n}$ in the continued fraction expansion $\alpha=\left[a_{0}, a_{1}, a_{2}, \ldots\right]$ are bounded then $\alpha$ is badly approximable.

Give, with proof, an example of an irrational number which is not badly approximable.

[Standard facts about continued fractions may be used without proof provided they are stated clearly.]