I am working on a problem that related to rational approximation of real numbers.
I am looking for a bound of the form:
Given a positive real number, $\alpha \in (0,1)$, there exist positive integers, $p$ and $q$ such that $0< p \leq q$ and,
$$\left|\alpha - \frac{p}{q}\right|<f(q)$$
where $f(q)$ is some function of $q$.
I am almost certain that there are some good bounds given that there are lots of similar statements in the field of Diophantine approximation. Unfortunately, these bounds are for $\alpha \in \mathbb{R}$ and therefore do not necessarily that $p \leq q$. Intuitively speaking, however, it should follow that $\alpha \leq (0,1)$ implies that the rational approximation, $\frac{p}{q} \leq (0,1)$ which would then imply that $p \leq q$.