Let us consider the set $\ell_\neq$ of bounded sequences of unequal real terms. We use the following descriptions. A sequence $x=(x_0,x_1,...)\in\ell_\neq$ is dense if, for all $\varepsilon>0$, whenever $\inf x<\xi<\sup x$, there is $i\in\Bbb N$ such that $|x_i-\xi|<\varepsilon$. A sequence is rational if all its terms are rational. The separation of a sequence $x=(x_0,x_1,...)\in\ell_\neq$ is defined as $$\operatorname{sep}x:=\dfrac{\inf\{|i-j|\,|x_i-x_j|:i,\!j\in\Bbb N;\,i\neq j\}}{\sup\{|x_i-x_j|:i,\!j\in\Bbb N\}} ,$$ and we write $\ell_+:=\{x\in\ell_\neq:\operatorname{sep}x>0\}$.
A dense sequence in $\ell_+$ is known, but it is not rational. Also known is a sequence of maximal separation, which is rational but not dense. The following questions arise.
(Q1) Is there a dense rational sequence in $\ell_+$?
(Q2---more demanding) Can a sequence in $\ell_+$ enumerate the rationals in an interval?
Ultimately, I am looking for the supremal separation (and, if possible, a sequence attaining that supremum) of sequences in $\ell_+$ of the following types: (A) dense; (B) dense and rational (if such exists), as in Q1; and (C) enumerating the rationals in an interval (if such exists), as in Q2.
