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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?

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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 $\frac 1{(n-1)(n-2)}$

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