My excellent friend @realityminus3 asked:


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.


  • @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
  • @soupie66: Whittaker and Watson.

  • @Long_tailed_tit: Brannan might be too simple. But written by the @OpenUniversity and so high clarity of explanation.
  • @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 open-educational-resource 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 run-down 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!