Without criticizing any other people’s recommendations on this thread, I think it’s easy for people who have been doing proof-based math for a few years to recommend books that look clear and easy to them now, without remembering the time, effort and other support (e.g. great teacher and classmates) that may have been necessary to make use of that book a good experience.

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” :)