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I apologize if this question is inappropriate for this site, but I'm new here and am not entirely sure where to direct it.

I've just begun a course on real analysis and linear algebra at my university and it's the first time I've ever taken a course almost entirely devoted to proof. I'm finding the material very difficult to wrap my head around, specifically, the problem I'm having involves conceptualizing or visualizing the material to allow myself to begin a given problem. We are using the book call Tools of the Trade by Paul Sally and this book presents almost no practical guidance on approaching proofs or the material itself and neither does my instructor (either in class or in private).

Can anyone offer any advice or suggestions as to how I might better learn to internalize this information? If not, does anyone know of any good books that I might read to gain such insight? The material we have covered so far includes the construction of the integers, rationals, and reals (using Cauchy Sequences) along with a basic discussion of functions and set theory.

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  • $\begingroup$ Look for William Chen's lecture notes, and the books by the Trillia group. <ocw.mit.edu> has lots or courses (with lecture notes, homeworks, exams) on line. Wikipedia has a series of texbooks on a variety of topics. $\endgroup$
    – vonbrand
    Feb 3, 2013 at 4:37
  • $\begingroup$ If you want, I can add some recommendations on good analysis books too. $\endgroup$
    – Red Banana
    Apr 16, 2013 at 23:21
  • $\begingroup$ Please do, thank you. $\endgroup$
    – vrume21
    Apr 17, 2013 at 2:04
  • $\begingroup$ @bam54 Done. I hope it helps. $\endgroup$
    – Red Banana
    Apr 18, 2013 at 2:41
  • $\begingroup$ I don't know about others, but I found the book A First Course in Logic: An Introduction to Model Theory, Proof Theory, Computability, and Complexity very helpful in bridging this kind of gap. $\endgroup$
    – Adam
    Oct 11, 2013 at 20:55

3 Answers 3

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It is a good thing to try different books, in my experience as a self-learner I found that a lot of traditionally aclaimed books are incredibly hard, there's always an author that can help you to grasp core ideas easily, for example, in calculus I read a little of the calculus made easy by Silvanus Thompson.

Springer has a lot of titles on proofs, and there are also some books you should look:

Bridge to Abstract Mathematics: Mathematical Proof and Structures - Ronald P. Morash

  • This is a really nice book, it made a lot of things on set theory, logic and proofs a little easier to me.

How to Solve it - George Pólya

  • This is a classic book, I guess you must be aquainted with it.

HOW TO PROVE IT: A Structured Approach - Daniel J.Velleman

  • I'm about to read this one, it seems to have a nice purpose.

Linear Algebra As an Introduction to Abstract Mathematics - Isaiah Lankham, Bruno Nachtergaele & Anne Schilling

  • I dont remember how I found this book but perhaps it may be of help to your case,I found it in my library and it seems to be a mix of Linear Algebra and proofs, it seems nice for your case.

There's a class of books that may be also helpful for your case, the transitions to advanced mathematics:

Mathematical Proofs: A Transition to Advanced Mathematics - Gary Chartrand & Albert D. Polimeni & Ping Zhang

A Transition to Advanced Mathematics: A Survey Course - William Johnston & Alex M. McAllister

A Transition to Advanced Mathematics - Douglas Smith & Maurice Eggen & Richard St. Andre

Also some references on real analysis:

A First Course in Mathematical Analysis - David Alexander Brannan

  • I really loved this book, as the author says in the preface: Changes in the school curriculum over the last few decades have resulted in many students finding Analysis very difficult. The author believes that Analysis nowadays has an unjustified reputation for being hard, caused by the traditional university approach of providing students with a highly polished exposition in lectures and associated textbooks that make it impossible for the average learner to grasp the core ideas.

Introduction to Real Analysis - Robert G. Bartle & Donald R. Sherbert

  • This is also a good one, a little harder than the first one but still nice.

I was thinking about this answer and I reminded of one thing that I took a lot of time to understand: the concept of the best book. The best book is the one that makes you learn. In an analysis course, most people will tell you to read Rudin's book, for calculus they'll say you to read Apostol's book, this is kinda invalid and it really depends on your background in mathematics, it's good to remember that such books were written in different circumstances and that the authors presume that the readers know some things. I'm not discrediting these books, they're nice, but it will be much better if you learn with something easier and then try to read these hard books later. Always try to find books that are compatible with your mind, this will make your mathematical experience a lot better. You can also try to read topics spoken by different people: Having trouble with one author's definition of sequence? Try to read the definition by other author, I'm doing this with the books I mentioned: When something is hard on Sherbert's book I read what Brannan has to say about it.

I hope it helps.

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    $\begingroup$ The description given underneath Brannan's book describes exactly the problem I have with advanced math, particularly at my institution. It seems very relevant, your bringing up Rudin's text in comparison to the others above because I am continually confounded by it as a textbook. As an exposition of mathematical knowledge, it's fascinating and a work of great depth. But as an educational tool, I feel that one could not choose a worse book for a course functioning as an introduction to analysis. And yet, at many institutions, that is exactly the text used. $\endgroup$
    – vrume21
    Apr 18, 2013 at 3:01
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    $\begingroup$ This situation seems analogous to teaching someone to play a sport by having them compete with professional athletes: of course, something will be gained from the experience, but what will be lost? Certainly, this experience will be much more painful than simply taking an approach that focuses more completely on elevating the student to the level of the professional, rather than simply showing the student how professionals operate, often with the expectation that the student discover for himself how to bridge the gap. $\endgroup$
    – vrume21
    Apr 18, 2013 at 3:01
  • $\begingroup$ Yes. I don't know why, but there's a weird love for tradition, the idea that one universal formula was found (in this case, Rudin's book) and they never understand that the students could need something different. $\endgroup$
    – Red Banana
    Apr 18, 2013 at 3:46
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If it's the very idea of (fairly) rigorous proof that is bugging you, then can I warmly suggest looking at the excellent

Daniel J. Velleman, How to Prove it: A Structured Approach (CUP, 1994 and much reprinted, and now into a second edition).

From the blurb: "Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs." Which is, from what you say, exactly what you need.

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I am not sure if this is still relevant to you but I will make the following suggestions that have helped me have that click moment when it comes to proofs.

1) You need to realize how important definitions are. How can you prove a statement involving a concept that you do not fully grasp? It is essential to develop an intuition about the concepts: Why did we need to define X? Why is it relevant? Find examples that are concrete and familiar to you but also try to look for examples that feel "weird". As the next step you should try to feel very comfortable with the formal definition and be able to manipulate it easily ( for example, if it involves epsilons and deltas). You will find that this often comes easily if you have a good intuitive grasp of the concept.

2) Start with "follow your nose" type of proofs: If you have to prove a theorem and the only tool you have is a definition then often the proof will only involve manipulating a definition in a more or less straightforward way. After practising these for a while, they will come very naturally to you, no matter the area of mathematics you are studying.

3) Following an analogy from one of my lecturers : "Maths is like a beach filled with rock pools. Each rock-pool requires a different sneaky trick to cross it and some of the sneaky tricks that you learn crossing one rock-pool can be applied to others." I think it is important for you to actively read different kinds of proofs and figure out what the key ideas and tricks are. You will often find that there are standard ways to attack certain types of proofs, often one ingenious trick that seemed completely mysterious when you first saw it becomes a standard thing to try an will come naturally to you.

4) When trying to prove a statement first do a draft version. Look closely at what you have to show. How can you simplify the result you are trying to get? After you have simplified it , are there any theorems that seem appropriate? Look closely at the assumptions, play with the concepts and what you can deduce from the assumptions.A couple of strategies will come to you and often one of them will work! Try to get the general idea of the proof and only afterwards should you care about formalizing it and filling in the details.

5) Learn form proofs you read. Why would someone try a certain trick to attack this specific problem? What in the statement of the result prompts towards a given strategy or theorem? Even when things in proofs seem to come from nowhere and there are certain steps that you think you would have never thought of, it is very instructive to look hard for clues that would suggest trying those routes.If you really can't find anything either discuss it with someone or file that trick under those "sneaky tricks" that you can apply to other rock-poles. You will eventually find that when you first read a statement you are trying to prove you automatically jot down some possible strategies from prompts in the statement.

6) Be very careful not to assume things that might seem intuitively clear to you whereas in fact there is some weird counter-example that proves you wrong. This tends to happen when one is time pressured or frustrated so look out for that!

I hope some of this helps :).

P.S: I am an undergraduate so take this with a pinch a salt. Also I apologize if I have not expressed myself very well, English is not my first language and these kind of things are often hard to explain. If someone feels like they can express any idea better please feel free to edit my answer.

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