Books for... Real Analysis
My excellent friend @realityminus3 asked:
Hey mathematics twitterdo any of you know of a good undergraduate textbook for Real Analysis?
— RealityMinus3 (@RealityMinus3) December 21, 2020
tl;dr
What would I buy, having read all of the comments and thoughts?
I’d probably pick up Alcock and Cummings to build up my intuition, and then Rudin as a reference. I’d be tempted to download Thomson et al. as well, seeing as it’s free.
Book list
What follows is all of the books mentioned (in alphabetical order by [first] author), and then a lightly edited collation of the replies, including some shared elsewhere.
 Abbott: Understanding Analysis
 Alcock: How to Think About Analysis
 Apostol: Calculus (very expensive)
 Bear: Introduction to Mathematical Analysis
 Bear: A Primer of Lebesgue Integration (advanced)
 Binmore: Mathematical Analysis
 Brannan: A First Course in Mathematical Analysis
 Bressoud: A Radical Approach to Real Analysis
 Bryant: Yet Another Introduction To Analysis  also available on Cambridge core
 Burkill: A First Course in Mathematical Analysis
 Burn: Numbers and Functions
 Clapham: Introduction to Mathematical Analysis
 Cummings: Real Analysis
 Garling: A Course in Mathematical Analysis, Vol I
 Green: Sequences and Series
 Haggarty: Fundamentals of Mathematical Analysis
 Hart: A Guide to Analysis
 Marsden: Elementary Classical Analysis
 Reade: Introduction to Mathematical Analysis
 Ross: Elementary analysis
 Rudin: Principles of Mathematical Analysis (Baby Rudin)
 Spivak: Calculus (advanced)
 Stephenson: Mathematical Methods for Science Students
 Thomson, Bruckner and Bruckner: Elementary Real Analysis (free)
 Whitaker and Watson: A Course of Mathematical Analysis
Commentary

@BhaiTeraSakhtHa: Walter Rudin.

@alephJamesA: Rudin.

@tasminS: People will say Baby Rudin, and it is the classic but I think it’s more fun as a third time round text — for learning from I would say Abbott’s Understanding Analysis.

@Moon0nASpoon: +1 for Abbott, that’s what I taught from this summer and I really liked it. Very readable, and big emphasis on learning how to put proofs together, especially in the early chapters.
 @professorBrenda: Highly recommend “Understanding Analysis” by Stephen Abbott. I call it “the book so nice I used it twice” because I learned from the 1st edition while in undergrad, and now I teach my students using the 2nd edition
 @ilsmythe: Abbott is fantastic. I used it to teach an honors section of intro analysis at Rutgers (with the goal that they would be ready for Rudin the next semester) and I thought it worked extremely well.
 @benjamindickman: strongly second this choice. FWIW: if the goal ends up being to go further (I don’t think this will happen…) then i think HS Bear’s book is an accessible undergrad text on Real Analysis II
 @MrMansbridge: Spivak
 @themathdiva: That would be my choice

@soupie66 An amazing book to read before you even start is Lara Alcock’s fantastic book How to think about Analysis (OUP). I wish I had read it before and during my undergraduate course, for it is BRILL!

@Howat_Hazel: have forgotten anything I knew about Analysis but I know Lara Alcock writes excellent books

@ChrisBMaths: Mary Hart’s book, A Guide to Analysis
 @DarrenBrumby: Victor Bryant’s Yet Another Introduction to Analysis
 @MathematicalA: Available on Cambridge core

@soupie66: Whittaker and Watson.
 @Long_tailed_tit: Brannan might be too simple. But written by the
@OpenUniversity and so high clarity of explanation.
 @MathematicalA: Seems to be freely available on the internet

@Mathemacricket: My A Level teacher lent me this book before I started my degree: Fundamentals of Mathematical Analysis by Rod Haggarty.

@isleofmandan: Fundamentals of Mathematical Analysis by Rod Haggarty is quite accessible.

@sumsgenius: The maths dept at uni asked us all to work through Stephenson before we started the degree. It was a long time ago, though.

@JoeHarrisUK: some options from the Cambridge schedules

@matthematician: For accessibility and openeducationalresource availability: Thomson, Bruckner, Bruckner.

@darthkiks: To get a better feel for the motivation behind the theorems, I highly recommend “A radical approach to real analysis” by D. Bressoud
 @TChihMaths: I think I’ll let @LongFormMath plug his own book!
 @LongFormMath: (Cummings: Real Analysis)  It’s like Abbott’s book on steroids!

@profgoat: Now this (Bear: Lebesgue) isn’t a beginner book, but might be an advanced topic book, or a readable refresher for those of us 30 years out of our comps.

@profgoat: Marsden’s book is as old as dirt but I love it.
 @mathdocron: I liked “Elementary Analysis: The Theory of Calculus” by Kenneth Ross.
 @_qnlw: Just used it this year. The content and presentation are okay (some minor things I disagree with). But I am not terribly impressed by the choice and phrasing of the exercises.
I’ll leave the full rundown to Nicholas Jackson:
 @njj4: Hart is a good introductory book that covers sequences, series, continuity, limits, differentiation. Alcock is a very readable introduction that talks about how to think about the subject. Green and Clapham are nice little books. Burkill is a bit old now, and I never found it very readable  it was on the suggested list in my first year (1991) but we used Binmore instead, which was much clearer. Bryant is quite accessible. Burn is readable but strange  the proofs are broken down into guided exercises.
 @sam_holloway: I remember real analysis was the course I struggled to get a good textbook for (back in 1997/8). Bryant I’ve looked at since and it looked like the one that would have helped me!