Problem on Liouville's Theorem on approximation

Question

I'm unable to understand the red marked line. Can anyone help ?

Answer 1

We know that we can choose $p,q$ such that $|\alpha - p/q| \leq 1/q^2$ by Dirichlet's approximationtheorem (this works only if $\alpha$ is irratonial - that's the case here). Note that otherwise we already have $|\alpha - p/q| \geq 1/q^2 \geq 1/q^d$. By continuity of $f'$ we know $|f'(\xi)| \geq |f'(\alpha)|/2$ for $|\xi - \alpha|< \epsilon$ for some $\epsilon>0$. Increasing $C>0$ (we only have to check finite many $q$), we may suppose that already $|p/q-\alpha| < \epsilon$.