# A lower bound for continued fraction approximation.

It is known that, for a continued fraction expansion of an irrational $\alpha$ we have that:

$$\left| \alpha - \frac{p_n}{q_n} \right| = (\alpha_{n+1}q_n^2 + q_nq_{n-1})^{-1}$$

Show that the following lower bound for the error holds:

$$\left| \alpha - \frac{p_n}{q_n} \right| \geq \frac{a_{n+2}}{q_nq_{n+2}}$$