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Reddit mentions of The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs (Princeton Lifesaver Study Guides)

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We found 6 Reddit mentions of The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs (Princeton Lifesaver Study Guides). Here are the top ones.

The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs (Princeton Lifesaver Study Guides)
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Found 6 comments on The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs (Princeton Lifesaver Study Guides):

u/blaackholespace · 18 pointsr/math

> Mathematical Logic

It's not exactly Math Logic, just a bunch of techniques mathematicians use. Math Logic is an actual area of study. Similarly, actual Set Theory and Proof Theory are different from the small set of techniques that most mathematicians use.

Also, looks like you have chosen mostly old, but very popular books. While studying out of these books, keep looking for other books. Just because the book was once popular at a school, doesn't mean it is appropriate for your situation. Every year there are new (and quite frankly) pedagogically better books published. Look through them.

Here's how you find newer books. Go to Amazon. In the search field, choose "Books" and enter whatever term that interests you. Say, "mathematical proofs". Amazon will come up with a bunch of books. First, sort by relevance. That will give you an idea of what's currently popular. Check every single one of them. You'll find hidden jewels no one talks about. Then sort by publication date. That way you'll find newer books - some that haven't even been published yet. If you change the search term even slightly Amazon will come up with completely different batch of books. Also, search for books on Springer, Cambridge Press, MIT Press, MAA and the like. They usually house really cool new titles. Here are a couple of upcoming titles that might be of interest to you: An Illustrative Introduction to Modern Analysis by Katzourakis/Varvarouka, Understanding Topology by Shaun Ault. I bet these books will be far more pedagogically sound as compared to the dry-ass, boring compendium of facts like the books by Rudin.

If you want to learn how to do routine proofs, there are about one million titles out there. Also, note books titled Discrete Math are the best for learning how to do proofs. You get to learn techniques that are not covered in, say, How to Prove It by Velleman. My favorites are the books by Susanna Epp, Edward Scheinerman and Ralph Grimaldi. Also, note a lot of intro to proofs books cover much more than the bare minimum of How to Prove It by Velleman. For example, Math Proofs by Chartrand et al has sections about doing Analysis, Group Theory, Topology, Number Theory proofs. A lot of proof books do not cover proofs from Analysis, so lately a glut of new books that cover that area hit the market. For example, Intro to Proof Through Real Analysis by Madden/Aubrey, Analysis Lifesaver by Grinberg(Some of the reviewers are complaining that this book doesn't have enough material which is ridiculous because this book tackles some ugly topological stuff like compactness in the most general way head-on as opposed to most into Real Analysis books that simply shy away from it), Writing Proofs in Analysis by Kane, How to Think About Analysis by Alcock etc.

Here is a list of extremely gentle titles: Discovering Group Theory by Barnard/Neil, A Friendly Introduction to Group Theory by Nash, Abstract Algebra: A Student-Friendly Approach by the Dos Reis, Elementary Number Theory by Koshy, Undergraduate Topology: A Working Textbook by McClusckey/McMaster, Linear Algebra: Step by Step by Singh (This one is every bit as good as Axler, just a bit less pretentious, contains more examples and much more accessible), Analysis: With an Introduction to Proof by Lay, Vector Calculus, Linear Algebra, and Differential Forms by Hubbard & Hubbard, etc

This only scratches the surface of what's out there. For example, there are books dedicated to doing proofs in Computer Science(for example, Fundamental Proof Methods in Computer Science by Arkoudas/Musser, Practical Analysis of Algorithms by Vrajitorou/Knight, Probability and Computing by Mizenmacher/Upfal), Category Theory etc. The point is to keep looking. There's always something better just around the corner. You don't have to confine yourself to books someone(some people) declared the "it" book at some point in time.

Last, but not least, if you are poor, peruse Libgen.

u/horserenoir1 · 12 pointsr/todayilearned

Please, simply disregard everything below if the info is old news to you.

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Algebraic geometry requires the knowledge of commutative algebra which requires the knowledge of some basic abstract algebra (consists of vector spaces, groups, rings, modules and the whole nine yards). There are many books written on abstract algebra like those of Dummit&Foote, Artin, Herstein, Aluffi, Lang, Jacobson, Hungerford, MacLane/Birkhoff etc. There are a million much more elementary intros out there, though. Some of them are:

Discovering Group Theory: A Transition to Advanced Mathematics by Barnard/Neil

A Friendly Introduction to Group Theory by Nash

Abstract Algebra: A Student-Friendly Approach by the Dos Reis

Numbers and Symmetry: An Introduction to Algebra by Johnston/Richman

Rings and Factorization by Sharpe

Linear Algebra: Step by Step by Singh

As far as DE go, you probably want to see them done rigorously first. I think the books you are looking for are titled something along the lines of "Analysis on Manifolds". There are famous books on the subject by Sternberg, Spivak, Munkres etc. If you don't know basic real analysis, these books will be brutal. Some elementary analysis and topology books are:

Understanding Analysis by Abbot

The Real Analysis Lifesaver by Grinberg

A Course in Real Analysis by Mcdonald/Weiss

Analysis by Its History by Hirer/Wanner

Introductory Topology: Exercises and Solutions by Mortad

u/fgtrytgbfc · 11 pointsr/Thetruthishere

Pick up mathematics. Now if you have never done math past the high school and are an "average person" you probably cringed.

Math (an "actual kind") is nothing like the kind of shit you've seen back in grade school. To break into this incredible world all you need is to know math at the level of, say, 6th grade.

Intro to Math:

  1. Book of Proof by Richard Hammack. This free book will show/teach you how mathematicians think. There are other such books out there. For example,

u/sfsdfe556erg · 2 pointsr/learnmath

Even though most every proof technique (contradiction, induction, smallest counterexample etc) is shared by almost all branches of math, every branch of math has very specific goals or ways of going about proving statements. In your case you probably don't have time to learn how to prove statements in number theory, combinatorics or category theory or whatever so you must concentrate on analysis proofs. To that end, check out:

The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs by R Grinberg

Writing Proofs in Analysis by J. M. Kane

u/sunflux · 2 pointsr/UCSD

Hello, I think you're spot on about it making your life easier after struggling, and by taking this class and putting in the time, it will make other math courses much easier for you. Because of what you gain from the struggle, I would really recommend you take this over 142, if you have the time. I took 140A last fall, and although I only got a C, it took an immense amount of effort to even get that. The class is set up so that if you put in the hard work to understand the concepts, the homework, the proofs and so forth, you're gonna do well, and If you truly understand how to solve the homework problems, then the tests will be familiar (doesn't mean it will be easy).

Expect to put a lot of work in. This statement needs to be taken seriously for this class, I've talk to some people in the class who say they put in 40 hours a week. This is usually because the concepts do not come immediately and you have to constantly repeat and approach at different angles to find a good understanding.

I recommend having a supplementary text while you are studying from the dreaded Rudin. For 140A, you should be looking at compactness and chapter 2 very early on as this is a big hurdle in that class. Other concepts will be more familiar but still challenging.

​

Some recommended texts (definitely find your own that works for you)

https://www.amazon.com/Real-Analysis-Lifesaver-Understand-Princeton/dp/0691172935 (If you prefer "casual" explanations of the concepts, this help me survive chapter 2 of Rudin. There are useful book recommendations in the very back)

https://www.amazon.com/Elementary-Analysis-Calculus-Undergraduate-Mathematics/dp/1461462703 (Ross is used for the 142 series, and I find it is very helpful if you are struggling. If you are having trouble, start with the easier version of a problem and build up from there. The book mainly stays within the R\^2 metric, which is what makes it simpler)

https://minds.wisconsin.edu/handle/1793/67009 (at some point, you're gonna get stuck and you will have to look at the solutions. This is ok, but don't become reliant on it, that really hurt me in the end when I did that. Some of the questions are fuccckkkiiinngg hard, so when you hit that wall, take a look here. They give solutions that skips over a ton of steps, or might not be that good of a way to solve the problem, but this is a great resource)

https://www.math.ucla.edu/~tao/preprints/compactness.pdf (Who doesn't know who Terence Tao is? This is very helpful for giving an answer to "what is compactness used for?". It gives some intuition about what it is, and you should read it a couple times during 140A.)

​

So this is advice that I would give myself when entering the course, and maybe it won't apply to you. Since you got an A in 109 without too much trouble, you are definitely very ready for 140, and you have a very chance of succeeding. Stay curious, and don't stop at just the solution. Really question why it is true. You probably won't have this problem, but when it hits you (probably when you get to chapter 2) you have to keep at it and don't give up. Abuse office hours, ask lots of questions, study everyday etc. and you'll do well. If you want to get better at math then the pain is worth it.

u/utmostoftopmost · 1 pointr/math

What do you want to do, though? Is your goal to read math textbooks and later, maybe, math papers or is it for science/engineering? If it's the former, I'd simply ditch all that calc business and get started with "actual" math. There are about a million books designed to get you in the game. For one, try Book of Proof by Richard Hammack. It's free and designed to get your feet wet. Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand/Polimeni/Zhang is my favorite when it comes to books of this kind. You'll also pick up a lot of math from Discrete Math by Susanna Epp. These books assume no math background and will give you the coveted "math maturity".

There is also absolutely no shortage of subject books that will nurse you into maturity. For example, check out [The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs by Grinberg](https://www.amazon.com/Real-Analysis-Lifesaver-Understand-Princeton/dp/0691172935/ref=sr_1_1?ie=UTF8&qid=1486754571&sr=8-1&keywords=real+analysis+lifesaver() and Book of Abstract Algebra by Pinter. There's also Linear Algebra by Singh. It's roughly at the level of more famous LADR by Axler, but doesn't require you have done time with lower level LA book first. The reason I recommend this book is because every theorem/lemma/proposition is illustrated with a concrete example. Sort of uncommon in a proof based math book. Its only drawback is its solution manual. Some of its proofs are sloppy, messy. But there's mathstackexchange for that. In short, every subject of math has dozens and dozens of intro books designed to be as gentle as possible. Heck, these days even grad level subjects are ungrad-ized: The Lebesgue Integral for Undergraduates by Johnson. I am sure there are such books even on subjects like differential geometry and algebraic geometry. Basically, you have choice. Good Luck!