> You basically start out by constructing the reals from scratch Not necessarily -- a lot of books just take the existence of a unique set with certain propert

> You basically start out by constructing the reals from scratch Not necessarily -- a lot of books just take the existence of a unique set with certain properties as an axiom and call it R. The main topics of basic real analysis IMO are differentiation, integration, (uniform) continuity, compactness, convergence, etc.; how to construct the reals from the rationals is a side point at most. This could be personal bias as I just don't personally think that the exercise of constructing the real numbers is very interesting.