How to learn how to do math proofs?

[r4nd0m]

This year I'm a freshman at university - physics - and we are just starting with mathematical analysis. I don't find it that difficult, but my problem are proofs. They are not hard, but I sometimes can't prove even the easiest things (I know why it is so, but can't put it down on the paper). Can you recommend me a good way to learn how to prove? Are there any good online books available?

 

[HallsofIvy]

The best way to learn to do proofs is to read proofs. Go over the ones in your book in detail, observing how they go from one step to the next.
Next, look at the statement of a proof in your book, put the book aside and try it yourself. If you get stuck, go back to your book, read the first couple of lines of the proof and see if they are anything like yours (there are, of course, many different ways to prove anything). Keep doing that- trying to prove things yourself and comparing them to what the book has- you will improve.

 

[TD]

It will also help if you know the different types of proof. Although there are plenty, some are very often used. If you know the conceptual way of how those proofs work, it is easier to construct it yourself, to know what you are trying to get as a result.
For example, proving something by contradiction, by induction, ...

Check out this page, http://en.wikipedia.org/wiki/Mathematical_proof" , it also contains links to different types of proofs with examples

 

[mezarashi]

There's also some theory behind proofs. Or rather the methods of doing proofs for mathematicians as I have so read. Probably not an exhaustive list, but they are good starting points when you need to prove certain statements.

* Direct Proofs
* Proof by Contradiction and Reductio ad Absurdum
* The Contrapositive and Equivalent Forms
* Existence Proofs
* Uniqueness Proofs
* Mathematical Induction

Reference: http://www.math.csusb.edu/notes/proofs/pfnot/node4.html#SECTION00040000000000000000

 

[cscott]

 

[amcavoy]

You might also want to check http://www.artofproblemsolving.com" . There's a lot about proofs and related competitions.

Alex

 

[Icebreaker]

"What is Mathematics?" by Courant and "How to Solve It" by Polya are both good books that contain a number a proofs.

 

[robert Ihnot]

One way to start is to realize that there are a lot of proofs by induction that all work the same way. So after you do one, you are on your way to understanding. For example:

$$\sum_{j=1}^{j=n}j={\frac{n(n+1)}{2}$$

There are lots of similar sums, such as the sum of the squares from 1 to n, and the sum of the integers from 1 to n squared equal the sum of the cubes from 1 to n.