As mentioned in an early post I spent about nine hours cumulatively trying to construct the reals axiomatically from the bottom up using naive set theory. Though eventually I realized I made an error and tried fixing it, though this just made things more complicated and now I've spent several days doing this. Though now on the subject it got me thinking if I know the construction is unique up to isomorphism shouldn't I prove that as well? However continuing on with this fashion would bring me out to almost two weeks spent on the first few chapters of a text by Rudin.
Though I've sat in on about three intro to real analysis courses, taught by different instructors at a university and with the amount of time they each spend on some subjects e.g. at the start when discussing say completeness properties, right before jumping into metric spaces etc. my only conclusion is that essentially every undergraduate has taken much of these constructions on faith for the amount of time it seems required to do this in depth is no where near the few classes spent on the topic. Are people not expected to prove this stuff? How much time should I spend doing this? Its like each time I try to formalize something it opens up a new problem, when do I stop digging? Do I just accept this stuff like other students and move on to neighborhood Topologies etc. it seems were I actually enrolled in these classes I'd have no other alternative if I wanted to pass any exams.