TITLE: Sign in Dirichlet's approximation theorem
QUESTION [6 upvotes]: Fix $\alpha \in \mathbf{R}$. The classical Dirichlet's approximation theorem states there exist infinitely many rationals $p/q$ such that
$$
\left|\alpha-\frac{p}{q}\right|<\frac{1}{q^2}.
$$

Question. Fix $\alpha \in \mathbf{R}$. Is it true that there exist infinitely many rationals $p/q$ such that
  $$
0\le \alpha- \frac{p}{q}\ll\frac{1}{q^2}\,\,?
$$

REPLY [10 votes]: Yes, this follows from considering the continued fraction of $\alpha$. If $p_n/q_n$ is the $n$th convergent to $\alpha$ and $n$ is odd then
$$ 0\leq  \alpha - \frac{p_n}{q_n} \leq \frac{1}{q_n^2}.$$