Show that for a rational $a$ there is a finite amount of rationals $\frac{p}{q} \neq a$ such that $|a - \frac{p}{q}| < \frac{1}{q^2}$

Question

I want to shot that there for rational number $a \in \mathbb{Q}$, there is a finite amount of rational numbers $\frac{p}{q} \neq a$ such that $|a - \frac{p}{q}| < \frac{1}{q^2}$.

I know that for a irrational number $\alpha$, there are infinite such rationals, and it follows as a correlation from Dirichlet's approximation theorem.

However, I am not sure how it fails when $a$ is rational.

Help would be appreciated.

Answer 1

Let $a=m/n$. Then $$\left|a-\frac pq\right|=\left|\frac mn-\frac pq\right|=\frac{|mq-np|}{nq}.$$ Since this is a nonnegative integer over $nq$, it is either $0$ or $\geq \frac{1}{nq}$. Can you finish from here?