Proficiency in Mathematical Proofs

[Mathematicize]

Where does one attain the skill/ability (skill vs. ability is a good question in itself) to do proofs? By "do proofs" I mean prove existing theorems. For instance I am really no good with ε-δ proofs. The concept is so easy, but when I have to prove something using ε-δ I either have no idea what to do or I do something wrong.

I have heard that the only way to become good at proofs is to do tons of them. I have the idea that this is incorrect in the sense that perfect practice makes perfect not just lots of practice.

1. 
   So how does one perfectly practice writing proofs

I find doing proofs for class rather stressful since there is a deadline and a grade. What would one suggest for doing proofs in my free time as I way to gain proficiency? Additionally I am a decent problem solver. When it came to calculus questions and physics questions I could usually link the theorems or see the bigger picture to come up with the answer.

1. 
   Is there any recommended text that really changed your skills with proofs?

 

[lurflurf]

The skill/ability comes from practice. εδ proofs in particular require working with inequalities.
Try to practice proving simple things. Just make sure you understand each step, do not just push symbols around. Some I like
prove sqrt(3) is not rational
prove sqrt(3)+sqrt(7)+sqrt(17)+sqrt(19) is not rational
Prove sin(x),exp(x) and similar are not polyniomials
prove 17 is not even

I do not like how to proof it type books. Just read a nice book about and interesting subject. Calculus or linear algebra are ok; but algebra, discrete math, geometry, and number theory might be better. Proofs are problems, so a decent problem solver can become a decent prover. Imagine you are trying to convince someone (vey skeptical but logical) of the result, and they keep objecting, smash all possible objections.

 

[haruspex]

I am really no good with ε-δ proofs. The concept is so easy, but when I have to prove something using ε-δ I either have no idea what to do or I do something wrong.

In what way do you go wrong? Do you struggle to get the chain of reasoning right (i.e. the overall logical structure of the proof) or is it in finding the algorithm for choosing δ given ε, or maybe in manipulating the inequalities? Can you show an example?
In general, there are multiple skills: being clear about what constitutes a logical sequence; experience in different techniques (induction, reductio ad absurdum, counting arguments..) and when to try them; imagination; trying some simple cases to get insight; ...

 

[Stephen Tashi]

For epsilon-delta type proofs, it is worthwhile to study the proofs of important general results that involve such a technique - for example, proofs of the various cases of L'hospitals rule.

 

[MarneMath]

I think the way a person 'perfectly practices a proof' is by getting it wrong, then figuring out why it is wrong. Experience is the greatest teacher. You dive into a problem write what you believe is way you should go about it. Once you think it's write, present it to someone or look it up. If it's wrong, there is much value in figuring out why it is wrong. I've found that majority of whatever insight I have came from fixing my mistakes.

Books, just read what people tend to recommend, Aposto, Spivak, Courant, Rubin, Munkres, etc and you'll find plenty of opportunity to practice your proof skills and thanks to the internet, plently of people who know how to work through most of those problems.

 

[deluks917]

I learned the concept of a proof from Spivak's calculus. When I began reading the book I distinctly remember not understanding what a "proof was." However after doing most of the exercises in the first 3-4 chapters I basically understood. I had the same experience in relation to epsilon-delta proofs. I didn't understand them at all until I had done about 20 problems on them. I did not fully understand them until maybe a year or two later of seeing the concept.

 

[dkotschessaa]

I am fortunate in that my university has a "Bridge to Abstract Mathematics" course which is all about doing proofs, set theory, etc. A lot of people have to learn this stuff within the courses they are taking, sort of on the fly.

If you don't have a class like that, or even if you do, Daniel Vellaman's "How to Prove it" is an awesome, fantastic book. It's very friendly and approachable and readable, and has lots of examples and problems. (Get the later editions which have answers to selected problems. The earlier ones are lacking in solutions, which is annoying).

I did the first 3 or 4 chapters of this book over the summer in my own time, and it put me way ahead of the curve in Bridge.

With regards to ε-δ proofs... They are introduced usually in the first semester of calculus for some reason, before you have learned how to do proofs. They come out of nowhere and textbook authors insist on putting it there even though it confuses students and doesn't teach them anything about calculus that they are ready to process yet. You're going along learning calculus, and then here comes a bunch of greek letters and "there exists's" and "Such that's" and it's all very in your face. So don't worry about those quite yet any more then you might have to. If you *have to* do them, like on a test or something, just copy the damn examples from your textbook until you've pretty much memorized how they do it. In any other case this would be horrible advice for doing proofs, but sometimes you have to use brute force. :)

-Dave K

 

[bonfire09]

Eventually you do get better as you do more proofs. You see that a lot of them share similar patterns and they can be used to prove other propositions as well. But then there are those proofs that we struggle on and that are not so clear. This is where I am at. I am able to do 80 percent of all the problems without a hitch but there are those 20% that I am not able to get without looking up the solutions. I think that is where a lot us are at including me.

 

[Mathematicize]

Thanks for all the responses. I have gotten much better at ε-δ proofs. Last semester I aced the course and am even doing well with ε-δ proofs in the subsequent course :)