I know we define the reals as the equivalence classes of Cauchy sequences, but here the author is trying to motivate this and I'm not totally following how. Here is my summary of what he's saying. For context, right before this section, we proved that a Cauchy sequence is a convergent sequence.
Summary: First, he says that we should add to the rationals the limits of a Cauchy sequence of rationals which we know to be a real number. But, we don't know if this is enough yet for $\mathbb R$ since he didn't define $\mathbb R$ yet. So, he raises the possibility of Cauchy sequences of irrational numbers and ponders whether or not they converge to a "new variety of real numbers." It seems not because the Cauchy sequence of reals has the same limit as Cauchy sequence of rationals, and he gives a proof. So basically what we can conclude up to this point is that we should add to the rationals the limits of a Cauchy sequence of rationals (this accounts for the limits of a Cauchy sequence of irrational numbers, hence the reals in general, which is confusing bc we're trying to define the reals and how do we know the reals consist of the irrationals and rationals before defining it??)
Anyway, he says "Another informal demonstration that we should get every real number as a limit of Cauchy sequences of rational numbers is the infinite decimal expansion." Hold up, this is where I'm confused because "another" suggests that we already showed EVERY real number is a limit of Cauchy sequences of rational numbers, but instead we showed that every Cauchy sequence converges to a real number. I don't think these are the same. Moreover, the author didn't yet define at this point in the book that the reals = equivalence classes of Cauchy sequences. So I'm quite confused.
He then goes on to explain an actual explanation for "every real number is a limit of Cauchy sequences of rational numbers": if we assume a real number $x$ has an infinite decimal representation, then we can use that representation as a Cauchy sequence.
Can someone paraphrase what the author is saying as well as explain where the first explanation of "every real number is a limit of Cauchy sequences of rational numbers" is? I'm confused because this "intuitive"/"informal" discussion is just making me more confused.
Here's a link to the textbook: https://vdocuments.mx/strichartzthe-way-of-analysis-2000.html
It's page 33-34. I also attached photos below.