Or maybe they are just way smart than me? :)
Either way, when considering possible books to use, I would ask the following question - is there a chance this is “too easy” to use / read, while still claiming to be about analysis? (I.e. calculus books fail this test because they don’t say they are about analysis).
Then start with the easiest one unless there are really good reasons not to.
My own specific advice would be:
1) make sure you have had practice with proof based math before. If not, or you need the practice, get a copy of chartrand’s “introduction to mathematical proof” and do some exercises from the first 10 chapters. If you can do them easily, move onto analysis, if not, work through those 10 chapters first.
2) The book I personally like best for self-study is Abbott, “Understanding analysis” particularly if you can get the solutions manual, I think the explanations of the proofs are very good.
3) I would also recommend Lara Alcock’s book “How to think about analysis”, which is NOT a textbook, but has a lot of useful information and advice on how to learn analysis.
Also, obvious but worth repeating, if you are taking less than one hour per page to get through an analysis text, or don’t have pencil and paper in hand while going through the book, “ur doing it wrong” :)
I spent a decade with a terse text book on abstract algebra and went nowhere. It's for people who are either already enjoying concise math theorems or have the nack for it. I was missing a few bricks. A few years later some guy here or on reddit suggested a book that is vastly simpler, so simple it felt like HS but it cleared a few misunderstandings I had about notation and meaning. All of a sudden that 5$ ebay book had more value that my 100$ old paperbrick.
As in my posts in this thread, before real analysis should be about three books in linear algebra, and that is also likely an okay place to learn to read and write proofs -- a course in abstract algebra with a good teacher as I mentioned is easier, still.
Sadly it's a fact that the academic computer science community has too many chaired, full professors who got their education in mostly just computer science, had few or no good math theorem proving courses, in their current work try to get deep into math with theorems and proofs, but, alas, consistently make serious mistakes in notation, how to state theorems, how to write proofs, etc. I saw the same thing, sad to see, from a EE prof working hard in coding theory. It shows. Apparently a person can get competent reading and writing proofs in some early, appropriate pure math courses or not at all.
Yes, real analysis, advanced calculus, differential equations, differential geometry, mathematical statistics, stochastic processes, etc. are way too difficult as places to learn to read/write proofs. Similarly even for more advanced material in linear algebra.
I did quite like Bartle's "Introduction to Real Analysis" and "Elements of Real Analysis" books, kind of surprised that nobody else has brought them up. I think they strike an excellent balance between rigor and actually being comprehensible and approachable to people that aren't already familiar with proofs.
Real analysis is often the first class in a math bachelor's program where things really get far from intuition and much harder to wrap your head around than the material from earlier classes.
Making that the first proof-based class is just piling one hard thing on top of another.
I think we'd be better off if we made basic calculus proof-based, at least for people such as math majors who are going to need to learn to follow and do proof-based math at some point. For those going into fields where they will not need to read and do proves, have a separate "practical calculus" track.
You can start out with more informal proofs at the start of basic calculus, and slowly step up the level of rigor throughout the year.
One thing that helps enormously in math is to have to hand in exercises to check your understanding. Making these exercises together (over a few beers, for example) helps to have some confirmation, reference, and fun.
I have learned real analysis by using the provided lecture notes. They were maybe 200 pages and I think this works better to get a grasp on the subject than using a textbook. If you use a textbook with 500 pages, you probably gonna skip or forget 75%. Lecture notes can be better tailored towards the background of the students and the contents of the course.
Also, I think an hour per page is a little long, but it just depends on the density of the textbook you're using.
Last thing, good luck, try to find a partner to work on problems, and don't despair: you might feel stupid at times but analysis is just hard.