Reddit reviews How to Read and Do Proofs: An Introduction to Mathematical Thought Processes

You need to develop an "intuition" for proofs, in a crude sense.

I would suggest these books to do that:

Proof, Logic, and Conjecture: The Mathematician's Toolbox by Robert Wolf. This was the book I used for my own proof class at Stony Brook - (edit: when I was a student.) This book goes down to the logic level. It is superbly well written and was of an immense use to me. It's one of those books I've actually re-read entirely, in a very Wax-on Wax-off Mr. Miyagi type way.

How to Read and Do Proofs by Daniel Slow. I bought this little book for my own self study. Slow wrote a really excellent, really concise, "this is how you do a proof" book. Teaching you when to look to try a certain technique of proof before another. This little book is a quick way to answer your TL:DR.

How to Solve it by G. Polya is a classic text in mathematical thinking. Another one I bought for personal collection.

Mathematics and Plausible Reasoning, Vol 1 and Mathematics and Plausible Reasoning, Vol 2 also by G. Polya, and equally classic, are two other books on my shelf of "proof and mathematical thinking."

I heartily recommend Solow's How to Read and Do Proofs which would serve as a good foundation for thinking about math.

Absolutely! There is a second book I would suggest to fellow proofy people. This one is aimed directly at proofs. These two combined were invaluable to me:

How to read and do proofs

> Is there any good book with problems/examples that I could work through in order to thoroughly prepare myself to be able to write proofs for a Real Analysis I course?

Besides Velleman's "How to Prove it," try Mathematical Proofs: A Transition to Advanced Mathematics or maybe How to Read and Do Proofs: An Introduction to Mathematical Thought Processes.

The book I used in my "Intro to Proofs" course was A Transition to Advanced Mathematics. It was pretty good, but the edition that I used had several mistakes in it. Also, it's waaaay too expensive.

Now for the unpleasantries —

Suggestions aside, the main problem here is your "thoroughly prepare myself to be able to write proofs for Real Analysis" goal. Working through a proofs book on your own will be seriously challenging, but the thought of taking Real Analysis without at least two other proofs courses under your belt is terrifying to me. I had to take "An intro to mathematical proofs" followed almost immediately by a proof-based Linear Algebra course before I was even allowed to contemplate a Real Analysis course.

Come to think of it, how in the hell are you even allowed to do this if you haven't taken a proofs course before? Are you sure this is even possible? Are prerequisites not enforced at your school? No one, and I mean no one was permitted to take Abstract Algebra or Real Analysis without the required prerequisites at my university. The only way you could get around it was by being the next Andrew Wiles.

Just to drive all this home, I was a straight-A Physics/Math major with the exception of two courses: Thermodynamics and my first proofs course. I've never worked so damn hard for a B in my life. Come to think of it, I actually recall quantum mechanics being easier than my proofs course.

I'm being sincere when I ask you to reconsider this plan. You are asking for a world of pain followed by the very real possibility of failure if you do this.

TL;DR: Unless you are remarkably sharp and have loads of time on your hands, this is probably a mistake. You should take a more elementary proofs course before tackling Real Analysis. Good luck, whatever you choose to do.

Certainly recommend getting a copy of Rosenlicht if you're in this situation. This is what I'm currently reading in preparation for an honors-level analysis class. Should be pretty easy to follow even without much of a strong proof-based background.

As far as proof mechanics and mathematical foundations go, I recommend "How to Read and Do Proofs". It's not a bad reference to have at your side, just in case.