<s>[INST] How to contruct such a sequence of rational? How to order all  rational numbers from $(0,1)$ in a sequence $(x_n)_{n=1}^\infty$ in such a way that $$|x_n-x_k| \geq \frac{1}{(n+1)^2}$$ for $k<n$  ?
 [/INST] It seems that $\dfrac{1}{(n+1)^2}$ can be improved. If we do the first thing that comes to our mind:
$$
\frac{1}{2},\,\,\,\frac{1}{3},\frac{2}{3},\,\,\,\frac{1}{4},\frac{3}{4},\,\,\,
\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5},\,\,\,
\frac{1}{6},\frac{5}{6},\,\,\,\frac{1}{7},\frac{2}{7},\frac{3}{7},\frac{4}{7},
\frac{5}{7},\frac{6}{7},\ldots
$$
then $x_n$ is a fraction with denominator at $n$, and if $k<n$, then $x_k$ has numerator either $n+1$ or less than $n+1$. If its numerator is $n+1$, then $|x_k-x_n|\ge\frac{1}{n+1}$, whereas if it is less than $n$, then
$$
|x_n-x_k|\ge\frac{1}{n(n+1)}.
$$ 
</s>