# Difference between rationals in a certain set is at least a certain amount

Define $$A = \{\frac{p}{q} \in \mathbb{Q} \mid q \in \mathbb{N}, q < n, gcd(p,q) = 1\}$$. I am trying to prove that the difference of any 2 distinct elements of this set is greater than $$\frac{1}{n}$$. I have tried everything, starting from taking 2 arbitrary fractions in this set and just inequality bashing. Is there any succinct way to go about this?

It is not true. Consider $$\frac 14$$ and $$\frac 15$$. The distance between them is $$\frac 1{20}$$, which is less than $$\frac 16$$
If the two denominators are strictly less than $$n$$, the distance is at least $$\frac 1{(n-1)(n-2)}$$