serkan said:
problem is, I'm not confident about my ability to prove. even simple proofs include some tricks that i think i won't easily come up with at the moment. what would you recommend?
Not all proofs are created equal. Some are so simple or obvious that it's not clear that they should in fact be called proofs. Things like proofs that (x + y)^2 = x^2 + 2xy + y^2. On the other end of the spectrum, there are proofs that seem to require infinite genius to have found. The proof that no rational number's square is two, for instance... the contradiction in the proof is just so damn subtle!
In a single topic in math, you'll notice that many proofs follow a similar pattern or employ a similar "trick". Many mathematicians make their careers off of becoming the first or best at exploiting some kind of mathematical trick. For example, in set theory, Cantor invented the "diagonalization" trick to show relationships between the sizes of sets. The same trick can be used over and over in different ways. You can use it to show that the reals outnumber the rationals. But you can also use it in contexts of computability and formal logic to show that the number of truths and functions outnumber provable truths and computable functions.
By studying proofs, you become more familiar with these tricks. If you study proofs in Point-Set Topology, you'll become much better at proofs in point-set topology. If you come across a theorem which you've never seen, but you recognize topological elements of the problem, you'll have a clue that you should begin looking for ways to reduce the problem to a statement about homeomorphisms, compactness, connectedness, and continuous functions.
As a student, most proofs you'll be expected to exhibit on a test are going to be fairly easy ones. The ones you're most likely to encounter are ones that are very closely related to a definition of some sort. At my school, linear algebra was the course used to introduce students to proofs. Proofs on the test were things like "Prove that the operation 'rotate a vector by 45 degrees' for R^2 is a linear operator" or "prove that a nullspace is a linear space." These kinds of proofs you should be able to do (in any subject) with a small bit of studying.
Sometimes, though, if a major theorem's proof is presented in class, your prof. may want you to reproduce it, or to prove a similar theorem. For example, I had a class where the prof. taught us the proof for the irrationality of the square root of 2, then on the test, asked for the proof for the irrationality of the square root of 3. But if you knew the first (and actually understood how it worked), the second is really easy.
When you're working on a proof which is neither obvious nor has been covered in your class, that's where you're doing real mathematics =-) There is no clear cut path how to solve a proof in general, but as you learn more, you'll pick up lots of useful techniques.
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))))), Landau, turbulence, Reinolds number, Klimontovich and Mandelstam, first explaining alfa decay through tunneling (remember Gamow?),...
