What book to use for a first timer trying to learn real analysis?

[STEMucator]

Hey, I recently just finished my calc 2 course ( All my exams actually :) ), and I'm thinking about learning real analysis over the summer. Just to stay keen and it's a topic I'm really interested in.

I've heard Rudin is a staple ( heard it was tough as well ), but I would like to hear other recommendations before I embark on the hardest course ever ( Maybe not, but I think it's a big jump from calc 2 ).

 

[micromass]

Please stay away of Rudin. The book is more about Rudin showing how smart he is than about Rudin teaching analysis. Furthermore, there are some beautiful mathematical things that he totally murders (Lebesgue integration, multiple integration).
If you want a book that shows you just the facts and some (very elegant, but useless) way to derive them. Then Rudin is for you. There are books out there that also give the motivation.

Before we give recommendations, I think you should tell us how comfortable you are with rigorous calculus. I don't mean things like calculating integrals, you won't need that.

How well do you know the following:

1) Epsilon-delta definition of continuity.
Have you heard of it before? Did you do some basic examples? Can you show right now (without looking anything up) that $f:\mathbb{R}\rightarrow \mathbb{R}:x\rightarrow x^2$ is continuous?

2) Epsilon-"delta" definition of sequences
Have you heard of it before? Can you prove things like $x_n = 1/n^2$ is convergent? Did you ever encounter the notion of a Cauchy sequence?

3) Series
Did you understand this well? Did you ever prove anything? Can you prove that $\sum 1/n$ is divergent?

4) Derivatives
If I ask you to give me a function whose derivative is discontinuous. Can you give me one (without looking it up)?

5) Integrals
Did you ever learn the rigorous definition of a Riemann integral (this involves epsilons)? Did you prove the fundamental theorem of calculus?

6) Proofs
How good are you with proofs? If I ask you to show that a natural number is even if and only if its square is even, could you do it?

 

[micromass]

Also, is this going to be a self-study of real analysis? Or is this going to be in a formal class setting?

 

[STEMucator]

Also, is this going to be a self-study of real analysis? Or is this going to be in a formal class setting?

This will just be for my own personal hobby. I really just want to learn the subject itself.

1) Yes, for all epsilon I can choose $\delta = min\{ 1, \epsilon/5 \}$ etc...

2) A sequence converges iff it is cauchy. $|x_m - x_n| < \epsilon$ ...

3) Series were a strong point for me.

4) $f(x) = x^2sin(1/x)$ if x is not 0 and $f(x) = 0$ if x is zero.

5) Yes

6) Assume $a = 2n$, then $a^2 = 4n^2$ Since $2|4$, 4 is even. Since $n^2$ is a strictly positive natural number, $4n^2$ is also even. Hence $a^2$ is even.

Assume $a^2 = 2n$, then $a^4 = 4n^2$ ... having a bit of trouble going the other way.

EDIT : Perhaps this is not my strongest proof attempt, but I can with time write a good proof for a large portion of questions I do.

EDIT 2 : The point i want to make is I'm comfortable with rigorous calculus. All the proofs I know were done with epsilons for the most part. My prof has been teaching calculus/analysis for 40 years I believe ( hoping he'll be teaching real analysis in the fall ). I literally never had to look at 'Advanced Calculus by Taylor' even once, except to do some assignment problems twice. He somehow communicated the information in such a way that it was easy to understand, but he would prove everything through using epsilons.

For some reason though, calculus in general is easy to understand. It's the only topic that actually keeps my attention span going. I find whenever I go to class, I can keep up and always know what's going on and whenever the prof asks a question, I always have the answer ( okay 90% of the time, we all make mistakes sometimes ). Usually I finish writing the proofs to some of the theorems before my prof even finishes writing them on the board for the class ( especially the ones involving series & sequences as well as series & sequences of functions ).

I feel comfortable manipulating expressions with the methods of calculus and I think I have a very deep understanding of the value of its theorems ( even though we use those theorems as stepping stones to the next ). Fundamentally I feel I understand limits and neighborhoods very well and I know some basic things about open sets, closed sets, interior points and the interior set, boundary points, B(S), limit points, cluster points, convex sets, limsup/liminf and some real number axioms ( like the lub axiom ).

I also think I know some measure theory, but not yet what it's supposed to be used for. My prof has referred to it as Outer content ( Or outer measure I believe as i did some external research ).

 

[micromass]

One very beautiful book is written by Carothers: https://www.amazon.com/dp/0521497566/?tag=pfamazon01-20
It's one of my absolute favorite books on analysis because the author motivates everything. It's not a collection of theorems and proofs. Those are present however, but they are all tied together with a text that is quite motivational and historic.
The exercises in the text are plenty and they are all very good. If you were good in rigorous calculus, then you should be able to do this book.

Some minor points about the text seem to be that not every traditional topic in real analysis is done very extensively. This book absolutely takes the approach of function spaces (which is my favorite approach). But you should probably complement this book with another book some day.

Another nice book would be Knapp: https://www.amazon.com/dp/0817632506/?tag=pfamazon01-20 He's a very good writer and his books are very beautiful. He wrote 4 excellent books: basic and advanced analysis, and basic and advanced algebra.

You might want to read through these books, I highly recommend them.

 

[STEMucator]

One very beautiful book is written by Carothers: https://www.amazon.com/dp/0521497566/?tag=pfamazon01-20
It's one of my absolute favorite books on analysis because the author motivates everything. It's not a collection of theorems and proofs. Those are present however, but they are all tied together with a text that is quite motivational and historic.
The exercises in the text are plenty and they are all very good. If you were good in rigorous calculus, then you should be able to do this book.

Some minor points about the text seem to be that not every traditional topic in real analysis is done very extensively. This book absolutely takes the approach of function spaces (which is my favorite approach). But you should probably complement this book with another book some day.

Another nice book would be Knapp: https://www.amazon.com/dp/0817632506/?tag=pfamazon01-20 He's a very good writer and his books are very beautiful. He wrote 4 excellent books: basic and advanced analysis, and basic and advanced algebra.

You might want to read through these books, I highly recommend them.

Carothers looks like it's going to be an amazing read. Got my hands on a copy of it and I've been looking through the topics, it definitely is going to keep me busy.

I might have a read through of knapp afterwards followed by rudin at the end ( Just for the sake of completeness right :) )?

 

[micromass]

Carothers looks like it's going to be an amazing read. Got my hands on a copy of it and I've been looking through the topics, it definitely is going to keep me busy.

I might have a read through of knapp afterwards followed by rudin at the end ( Just for the sake of completeness right :) )?

Carothers and then Knapp is a nice idea! If you really want to see what's in Rudin, then you can do that too of course. In fact, you can probably read Rudin right now! You seem to have all the prerequisites for it. You should check it out. I think it's a horrible book, but there are many who disagree with me. A math book is a very personal choice, some people will like another style than other people.

 

[A. Bahat]

While I also think Rudin's exposition is pretty poor from a pedagogical perspective, many of the problems are quite good. It's a good book to have, but maybe not the best to learn from. Since you seem pretty comfortable with rigorous calculus, I think you wouldn't have trouble with Rudin.

Just please stay away from the chapter on Stokes' theorem. Seriously.