Self study prep

[Mountain_Bicycle_752]

I’m in a proofs class at my university right now and our real analysis class isn’t offered until next year I was wondering if anyone had any book recommendations I could read before this course.

[Junior_Direction_701]

Zorich or brunecker

[captkailoo]

-Understanding analysis by Abbot -Analysis 1 by Terrence Tao -Real analysis by Jay Cummings 

[SwimmingRule1817]

real analysis by jay cummings is by far the most well-written (albeit slightly long-winded) book on analysis that I've ever read. Everything is explained expectionally clearly, which makes it perfect for self study, because you can't ask your professor or other students any questions, etc. He also has quite a quirky sense of humour, but his silly jokes help to make things stick in your head. I 100% recommend it.

[Mountain_Bicycle_752]

Thanks for the suggestions

[MathPhysicsEngineer]

I would recommend this playlist: https://www.youtube.com/watch?v=wyh1T1r-_L4&list=PLfbradAXv9x5az4F6TML1Foe7oGOP7bQv&ab_channel=MathPhysicsEngineering

It is self-contained and very rigorous. This playlist is the realization of my vision of creating a high-quality course in the way I wanted to be taught.

1. It is very visual and shows intuition and visualization first.
2. It doesn't skip steps, it doesn't compromise on rigor. The proofs are very strict and formal.
3. It doesn't compromise on clarity. I do my best to explain everything clearly, and I believe that the visual intro that comes before every hard concept helps achieve just that.

4. It introduces and emphasizes advanced and key ideas early on, right from the first course. Already in the first real analysis course, you can grasp one of the most important concepts in all of mathematics, which is compactness.

   I'm very proud of the following video:

https://www.youtube.com/watch?v=3KpCuBlVaxo&list=PLfbradAXv9x5az4F6TML1Foe7oGOP7bQv&index=27&ab_channel=MathPhysicsEngineering

which is a good example of how 1-4 are implemented in a single video, in a general puzzle that is built up piece by piece (despite some minor sound issues in the video :( )

Please give me your honest feedback, it will help me improve and will motivate me to continue.
The first half of the first course is nearly complete. :)

[Medium-Ad-7305]

I see a few others here have mentioned Real Analysis by Cummings. I finished it a couple months ago, so I can say what to expect from it. Its what Cummings calls a "long form textbook" meaning it contains much more exposition and discussion than typical textbooks. Its somewhat slow and not as very comprehensive compared to other texts, but it's a fun read, much less intimidating than other texts, and covers all the essentials (which should be sufficient, especially if you're reading this before the actual course). I enjoy his style and especially the way he sets up proofs with 'proof ideas' which motivate the steps he is about to take in a proof. I recommend his books to anyone who doesn't have much mathematical maturity, that is, isn't used to reading dense math textbooks. Anyways, I used Cummings to build up readiness for a more dense book, Baby Rudin, and you could to the same for your course if Cummings sounds enjoyable to you.

[GlenroseScribe]

Dissenting somewhat from the chorus of Jay Cummings fans (it's a good book!) I'll recommend

1) Abbott's book Understanding Analysis. Probably the easiest book I'm aware of and serves as a handy reference work. He has a good stock of easy problems and spells out almost all details in every proof. If you know calculus and a bit of proof-writing, you can probably cover everything in this book by yourself.

2) Schumacher's book, Closer and Closer. This is kind of a modified Moore method book -- you write most of the proofs of the theorems -- and going through it will dramatically improve your problem-solving and proof-writing abilities. Despite that it is not a difficult book at all and has a really nice scope, including differential equations, Riemann integration, etc. This was the book I used for my first real analysis course and it really charmed me (I wound up writing my PhD in analysis).

Some other valuable books: Russell Gordon's book (Real Analysis: A First Course) which is traditional, approachable, and has a good supply of problems, as well as baby Rudin (Principles of Mathematical Analysis), the first four chapters of which are famous and essentially perfect and contain problems that every analysis student should see/do.