So I've gotten myself confused over a seeming tautology one uses when constructing the real numbers as equivalence classes of Cauchy sequences of rational numbers. Having constructed the real numbers in this way, once one makes sense of the term "positive real number" it is easy to extend the absolute value and in particular get a notion of convergence.
If one has a Cauchy sequence of rational numbers which represents the class of some real number, surely this sequence converges to this number, but how does one actually see this without begging the question?