Reddit mentions of Real Mathematical Analysis (Undergraduate Texts in Mathematics)

By the end of reading the books you should be able to do all the problems, but I don't think that means doing literally every single problem before moving on. I'd google courses that use the books you're reading and try their example problems. If you try to read on and feel they weren't enough then you can always go back and do more.

Also, going to Rudin for self-study seems mildly sadistic. For future reference Pugh's book Real Mathematical Analysis also has a huge helping of fantastic exercises while also being written in a way that's trying to teach you and not impress you. You should be capable of Rudin by the end of a real analysis course, but Rudin itself is not necessarily the best way to get there.

You are missing Abstract Algebra that usually comes before or after Real Analysis. As for that 4chan post, Rudin's book will hand anyone their ass if they havent seen proofs and dont have a proper foundation (Logic/Proofs/Sets/Functions). Transition to Higher Math courses usually cover such matters. Covering Rudin in 4 months is a stretch. It has to be the toughest intro to Real Analysis. There are tons of easier going alternatives:

Real Mathematical Analysis by Charles Pugh

Understanding Analysis by Stephen Abbot

A Primer of Real Functions by Ralph Boas

Yet Another Introduction to Analysis

Elementary Analysis: The Theory of Calculus

Real Analysis: A Constructive Approach

Introduction to Topology and Modern Analysis by George F. Simmons

...and tons more.

Sixty bucks!? Thirty bucks for Pugh and Rudin.


Hardy was a number theorist, but this book is straight analysis (I'm not sure how you approach analysis with number theory). From what I've seen, Hardy is very verbose and spends a lot of time on material you don't need to see the first time around. He also uses a lot of outdated terminology. Lastly, this book is calculus and analysis together. Presuming you've done calculus, you want to get straight to the analysis part. That's where the set theory and topology come in. "Modern" analysis (still pretty old) works in more general spaces and uses topological and set-theoretic ideas. It's actually very natural, and you'll wonder how you ever worked without them. You won't see this important modern presentation in Hardy, so you'd really be missing out. I'd buy Pugh/Rudin (or something easier) and use Hardy to supplement them, rather than the other way 'round.

A much, much more inexpensive copy with the same content is also available.

Rudin is definitely the classic, but for a more contemporary and "friendlier" (but no less rigorous) introduction to real analysis, some people prefer the book by Pugh.

Edit: The two books cover pretty much the same material in the same order. I've heard Pugh described as "Rudin, with pictures"

My school does a one semester intro using Understand Analysis and then a year long sequence using Rudin. I've been reading Real Mathematical Analysis and Pugh and I have to say that I am really enjoying it. Chapter two goes into more depth on topology that Rudin does in his book. There is also a lot of pictures and I am a visual learner.

I would just dive into it to see if it makes more sense! Here is a guide about delta epsilon proofs, which is one of the most common basic proofs you learn about in pure mathematics. Real Mathematical Analysis is a great textbook about real analysis. Also, if you're worried about the math, I would look into philosophical logic—Logic by Hodges is a good text for that and it won't involve any necessary background in math.