How to self study real analysis

Question

I'm a high school student, who is interested in learning real analysis, but it seems like every textbook I take doesn't really provide a way to intuitively think about the concepts. I understand it to some extent but I'm not satisfied with the level of intuition provided so far. Can someone help me recommend and textbook or a way to study it

Answer 1

It seems you need a book that guides you as a real lecturer, so any suggestion to the super terse Rudin's Principles of Mathematical Analysis is to avoid. I would recommend two books: the former is to fill any gap from calculus to real analysis, and the other one to have a light introduction to real analysis.

• Calculus: Early Transcendentals, by Bernard Gillett, Lyle Cochran, and William L. Briggs
• Introduction to Analysis, by Edward D. Gaughan

If you couple them together, you will be able to have a smooth transition from high school level math to (introductory) undergraduate math classes.

Answer 2

"Calculus" Volumes 1 and 2, 2nd Edition 1966 : Tom Apostol

Recommended to me as an alleged jewel. It is.

Answer 3

"Real Analysis: A Long-Form Mathematics Textbook" By Jay Cummings is where I recommend students coming out of high school/ finished A Level begin their studies on real analysis. It contains more intuitive explanations than any other calculus book I have encountered; it contains large, clear diagrams and the text is in a clear, readable font. I just think it's a good book for calculus newbies. It is lengthy (about $400$ pages), but I don't think this matters: Rudin's Principles of Mathematics took me much longer to get through as it is comparatively terse, although it's very good also (especially the exercises).

In conclusion, I would go through Cummings book first and then Rudin after for more of a challenge, or try them both at the same time. There are, of course, many other good calculus books, as mentioned in other answers.