Real Analysis $\mathbb{R}$e-view

On going through a layperson-readable take on Skolem's paradox here I noticed that the author (Steven Landsburg) talked about producing different models of $\mathbb{R}$ from different 'models' of $\mathcal{P}(\mathbb{Q})$ (powersets of the rationals). Why didn't he talk about Cauchy sequences? (because that's how we construct $\mathbb{R}$, right?)
As long as we don't mind throwing out the really boring sequences (like $\{1,1,1,1...\}$) with only finitely many unique elements, no emphasis on sequences is necessary. Because $\mathbb{Q}$ is countable, any subset is at most countable, and can be ordered and considered as a sequence. Furthermore, actually assigning an ordering is unnecessary...

Call a subset $S \subset \mathbb{Q}$ a Cauchy subset iif
1) $S$ is infinite
2) $\, \forall \, \epsilon > 0 \, \exists \, x \in S$ such that $\{ y \in S : |x-y|\geq \epsilon \}$ is finite

As an exercise, show (because my probably-nonexistent readership loves exercises, right?)
-$S$ is a Cauchy sequence under a particular ordering $\Leftrightarrow$ $S$ is a Cauchy set $\Leftrightarrow$ $S$ is a Cauchy sequence under every ordering.
-All the above statements/definitions work equally well with subsets of $\mathbb{R}$, not just $\mathbb{Q}$. (hint: show that condition 2 forces $S$ to be at most countable)