Wednesday, February 6, 2008

Having a hard time with proofs

Since making free PDF copies of our real analysis texts on our site we get an occasional appeal for help from users of the texts.

Here is the letter to us and my response.

R---- writes:

I happened across your book while doing some research on textbooks that could help me learn real analysis.

I've found that a lot of the topics I run into while working in AI and finance requires an understanding of real analysis. I've taken a significant amount of mathematics, statistics, differential equations, MVC, and linear algebra.

However, I'm finding myself getting stuck in ruts while trying to prove a lot of the exercises in the first chapter! Needless to say, it's quite frustrating when you can obviously see that it's true but can't formulate it into a proof. I realize that part of the battle to becoming good at math is being able to formulate the proofs on your own.

I was wondering if there were any hints or tips or books that I could read to help me with my inexperience in creating proofs.

Lastly, I greatly thank you for putting your book up online. I don't think I would have found it otherwise.

Thanks, R-------


My response:

Hi R------,
Don't be discouraged. I would say 95% of my students over the years would have said the same thing about proofs.

The Elementary Real Analysis text has an appendix where we talk about this. Certainly go there to be sure you know about Indirect Proofs, Direct Proofs, Contraposition and Proofs by Induction.

I would often for an exam announce in advance that a full proof would be required and tell the students explicitly which theorem. Even then many failed. So a first step is
to pick a theorem you like and memorize the proof and present that proof to as many people as are willing to listen. [I think of it like a jazz solo. If you want to learn how to improvise, first memorize someone's nice solo and perform it as often as you can. Your creativity won't follow until you learn first how to imitate.]

I am posting this on the alt.math.undergrad site and I'm hoping that others will have suggestions on how they managed to make this transition. Its way too many years back for me to remember how I managed, but I suspect it was because we studied Euclid in high school and had to produce hundreds of proofs in that style.


Anonymous said...

Hello there. I really like your suggestion. I am currently taking Real analysis class but I am having trouble understanding it. Now we are studying Sequences, just before Limit theorems. What theorem would you recommend me to memorize?

Brian Thomson said...

A good one might be proving that the limit of the sum of two sequences is the sum of the limits.

Loosely it uses the idea that if A is close to a and B is close to b then A+B is close to a+b. But that is called "hand waving" -- it is not mathematics. If you master the epsilon type proof and understand it you will be a long way towards getting the course. Memorize it and then teach in detail to anyone who is willing to listen. Become the class expert on this theorem!

Don't be discouraged that this doesn't come easily. The limit theorems are quite subtle and usually do take some time to absorb. Good luck...Brian