The starting point of this question is the:
IMO 1991, problem 6: Prove that for any $\alpha>1$, there exist a bounded real sequence $\{x_n\}_{n\in\mathbb{N}}$ such that, if $u,v$ are distinct natural numbers, $|x_u−x_v|\cdot|u−v|^\alpha>1$.
As shown by mercio in Inequality in a bounded real sequence, through the following corollary of the Hurwitz Theorem: $$ \forall r>\sqrt{5},\quad \#\left\{(p,q)\in\mathbb{N}_0^2:\left|\frac{1+\sqrt{5}}{2}-\frac{p}{q}\right|<\frac{1}{r\,q^2}\right\}<+\infty $$ the statement also holds for $\alpha=1$. Here comes my secondary question: if $S$ is the set of positive real sequences with the property $$\forall u,v\in\mathbb{N}^2,u\neq v,\quad \left|x_u-x_v\right|\cdot\left|u-v\right|>1, $$ what is $$ \inf_{s\in S}\,\left(\sup s\right) ?$$