Easy Real Analysis Books for Slow Learners

[PhizKid]

While attempting Rudin's Principles of Mathematical Analysis, I only got about as far as page 9 before losing him in the proof that $\mathbb{Q}$ is dense in $\mathbb{R}$. While his proof is only a few lines long, it does reveal some important properties that result from this theorem alongside the archimedean property which I find very insightful.

However, I needed a very long, in-depth, intricate explanation of the proof explained to me because I could not follow it, and was not able to derive the apparent corollaries that this proof revealed. Rudin used certain facts that I would not have been able to think of on my own no matter how many times I re-read his proof.

In an "easier" analysis book, it introduces these corollaries first, and then proves the density theorem afterwards (while Rudin does not mention these corollaries at all explicitly, so I assume he wants the reader to discover them on their own).

While I would prefer a presentation that includes excruciating details of proofs while assuming the reader has absolutely no abstract mathematical reasoning skill at all (or is not a very bright student in mathematics), I think too much hand-holding may be detrimental to math education so I am looking for a textbook that is somewhere in between the extremes of, say, Stewart's Calculus and Rudin's PMA (while one expects a bit much from the reader, the other assumes too little of the reader's knowledge, I guess).

Do you guys have any suggestions for introductory analysis textbooks for students that do not pick up concepts too quickly (something around the ballpark of at least being able to understand 9 pages of the textbook in less than 1.5 weeks, which is approximately how long I spent on the first 9 pages of Rudin and still didn't understand that particular proof).

 

[jbunniii]

You might consider checking out Spivak's Calculus. Despite the name, it is really an introductory real analysis book with very detailed and well-explained proofs. I would almost consider it a prerequisite before attempting to read Rudin.

 

[dkotschessaa]

Ah, you got Rudined! Congrats!

There actually IS one very helpful book that I found, which is the Springer test "Understanding Analysis"

Rudin is really a book for taking a class, having a professor explain it, and trying to fill in all of Rudins notorious holes. The Springer book is a good one for self teaching, though it only covers what you'd mostly get in a First semester class.

 

[dkotschessaa]

By the way, in my undergraduate Analysis class, we spent 2 weeks on the first 9 or 12 pages of Rudin. So that's actually not being slow. It is pretty normal for that book.

 

[verty]

Try this one if you like, it seems to start off slowly but does include metric spaces (in chapter 6) and a lot of n-variable stuff.

https://www.amazon.com/dp/0387974377/?tag=pfamazon01-20

Oh, I didn't read your whole opening post. You actually want a book that assumes no mathematical maturity? Is this a reaction to Rudin or do you really want that? I thought because you chose Rudin, you already had some proof ability, perhaps you are now having doubts?

I think you are somewhat confident in being able to handle some challenge. Therefore, JBunniii's suggestion of Spivak is good (quite a bit of challenge there, don't do all the questions) and you can try this Protter one as well, which has a larger scope than Spivak but may move more quickly.

Best of luck.

 

[CJ2116]

I've been attempting to work through "Elementary Real Analysis" for the past few weeks when I get some spare time and definitely recommend it! It gives very good explanations and has a lot of examples.

Best of all, the authors have it up for free on their website: http://classicalrealanalysis.info/com/Elementary-Real-Analysis.php

Although, it looks like the proof that Q is dense in R is about the same as the one on Rudin, so I'm not sure how much it will help on this.

 

[jbunniii]

I've been attempting to work through "Elementary Real Analysis" for the past few weeks when I get some spare time and definitely recommend it! It gives very good explanations and has a lot of examples.

Yes, this is a very nice book. A good choice if you want something maybe midway between Spivak and Rudin in terms of sophistication. (E.g., the authors cover topology on the real line and $\mathbb{R}^n$, whereas Spivak does no topology and Rudin does everything in metric spaces.) It also has good exercises: easier than either Spivak or Rudin, but still robust enough to ensure that you learn the material.

 

[Hercuflea]

My current professor wrote this book. There is an on-screen version, too but I think it might only be available to his in-class students.

https://www.amazon.com/dp/1107694043/?tag=pfamazon01-20