# Books for real analysis covering some specific topics

Book for real analysis:

The syllabus that needs to be covered is

• Elementary set theory, finite, countable and uncountable sets, Real number system as a complete ordered field, Archimedean property, supremum, infimum. Sequences and series, convergence, limsup, liminf.

• Bolzano Weierstrass theorem, Heine Borel theorem.

• Continuity, uniform continuity, differentiability, mean value theorem.

• Sequences and series of functions, uniform convergence.

• Riemann sums and Riemann integral, Improper Integrals.

• Monotonic functions, types of discontinuity, functions of bounded variation

I checked out the obvious recommendations given in the answers here which mainly include Rudin,Apostol,Pugh.

But the problem is none of them don't contain enough examples and material to learn the above topics on my own.

Having problem sets is a must have for the books.

I need a book that contains enough examples and good exercises(with hints possibly) that can help me learn the material and also help to prepare for the entrance exams that ask questions on real analysis.

Any help will be much useful.

• I'm surprised to hear anyone say that Apostol doesn't contain "enough examples and material". Carothers is another reference to consider. Friedman's "Foundations of Modern Analysis" is another I hear good things about. – Omnomnomnom Aug 29 '17 at 3:31
• @Omnomnomnom,I don't think Carother's contains topics on Improper Integrals or many of the topics that OP listed – Learnmore Aug 29 '17 at 3:38

Elementary Real Analysis by Thomson, Bruckner, Bruckner (2008).

http://classicalrealanalysis.info/com/Elementary-Real-Analysis.php

and a paper copy is only around \$34 or so.

It covers everything in your syllabus: Riemann integrals and improper integrals; uniform convergence; mean value theorem... everything on your list is there in the book.

I have used this book twice to teach introductory graduate real analysis (for master's-degree students; not the same as, but still similar to, a first-year Ph.D. course). The book has lots of discussion and explanation, and good exercise sets. There are hints for selected exercises in the back of the book.

Caveat, I'm not an analyst. I think it's a good book and it has good exercises but you can contact an analyst for an expert recommendation.

More generally, see AIM's Open Textbook Initiative, list of approved textbooks: https://aimath.org/textbooks/approved-textbooks/.

• Thank you very much;it will help me a lot – Learnmore Aug 30 '17 at 2:19