I am trying to do some estimates concerning continued fractions and according to my teacher I should get
$\left\lvert \alpha - \frac{p_n}{q_n}\right\lvert > \frac{1}{q_n+q_{n+1}} > \frac{1}{2q_{n+1}}$.
where $\frac{p_n}{q_n} = [a_0,a_1,...,a_n]$ is the continued fraction
The second inequality is clear, but I don't understand where I can derive the first one from. Thanks for any explanation.
I need this to prove the following: if $\sup_{n\in \mathbb{N}}\frac{a_n+1}{q_n^\sigma} < \infty$
then there exists $\gamma$ such that for all $(p, q) \in \mathbb{Z}\times\mathbb{Z}^* \left\lvert \alpha - \frac{p}{q}\right\lvert \geq \frac{\gamma}{2^{2+\sigma}}$
Thanks for any help.