> . I agree with your broader point that the structure and presentation of math can substantively change, even if the fundamental content doesn't at the undergraduate and graduate levels.
That was my main point. I wasn't intending to recommend intro analysis book. But now that you mentioned it,
for introduction to real analysis, there are even better textbooks out there. For bare-bones hand-holding stuff, there are "How to Think About Analysis" by Lara Alcock and "Real Analysis" by Jay Cummings. The former spends a lot of pages explaining the concepts of convergence and boundlessness and the latter, in addition to what Alcock does, introduces a bit of topology on the real line. Speaking of topology necessary for analysis, Rafi Grinberg's "Real Analysis Lifesaver" kicks ass. It tackles topology and convergence in the Euclidean Spaces (a leg up on the above-mentioned books). These extremely easy to read books (read in the given order) would easily prepare a dedicated student for tacking modern analysis.
If a student wants to see worked out example after worked out example (in extreme detail), Indian authors have you covered. Such STEM books seem to be a tradition there.
My favorite elementary analysis books by an Indian author are "Real Analysis" and "Sequences and Infinite Series" by N.P. Bali (No idea what the initials stand for; all I know is this author is Indian).
A book that serves up elementary analysis in bite sized portions and gives you only what's unavoidable (not simply necessary) is "Intro to Analysis" by Mikusinskis. It has no fat and no excess whatsoever. It aims to save you time as much as possible and wherever possible. Such style is usually reserved for grad level books.
Out of the three books you recommend, the ones by Tao and Pugh require an instructor nearby. If I recall correctly, Tao starts out by carefully laying down the basics of naturals in a way that's sure to leave most any math novice (especially those interested in analysis) bored and lost in the weeds. Pugh, on the other hand, is roughly at the level of, say, baby Rudin (if a bit less formal and austere) which makes it a VERY tall order for a beginner. At this point, the student might as well read Axler's book as it has more material, is more advanced and Axler is a phenomenal writer.
If a student is taught to fear and avoid anything that is "outside the scope of this class", "graduate level" etc, there's "Lebesgue Integral" by Johnston and "Friendly Approach to Functional Analysis" by Sasane. An average high school student can read these books :)
I own all these books.