There is a great 1st real analysis book that would work from HS level: "S. Abbott (2001). Understanding Analysis. Undergraduate Texts in Mathematics." (For comparison, Baby Rudin would be way more advanced than that, I'd schedule it even after Axler's Linear Algebra, which itself should go after a more matrix-y introduction to linear algebra, like Strang.)
Axler's Linear Algebra, which itself should go after a more matrix-y introduction to linear algebra, like Strang.)
Note however that this type of statement is strongly dependent on individual personality. For me, the correct order was definitely Axler first, then matrix-y later. (A position not to be confused, by the way, with "Axler only", which would indeed be a mistake, even for me.)
This will not be a long post; I have a simple question to ask: if you wanted to educate yourself to graduate level in mathematics, but didn't actually want to go to university, what would you do? I would ask for text-book recommendations, but I don't want to limit your responses (however, bear in mind that the wikipedia articles on, say, cardinality or well-ordering go over my head – they may skim my hairline, but over they go). Also bear in mind that while I personally have A-levels (British qualifications) in both Maths and Further Maths (which is to say, I know some calculus at least), there are probably plenty of people on lesswrong who don't and who desire the same information – so assume as much ignorance as you feel necessary (it's a shame, actually, that there isn't a sequence here on lesswrong for maths). What do you advise (if you think the query ill-defined, I would like to know that as well)?