Solution
= Solution
Because both fractions are reduced and $q>v$, they are distinct. Their difference is
$$
\left|\frac pq-\frac uv\right|
=\frac{|pv-qu|}{qv}.
$$
The numerator is a nonzero integer, so its absolute value is at least one. Therefore
$$
\left|\frac pq-\frac uv\right|
\geq\frac1{qv}>\boxed{\frac1{q^2}},
$$
where the strict inequality uses $v<q$. This is the <separation of distinct reduced fractions>.