Even at my college, where anyone taking analysis or algebra is expected to have taken that Abstract Vector Spaces class, the profs spend a pretty good deal of time going over how to write good proofs. So, hopefully the profs at your school will do the same. I don't know for certain, but I would imagine that most good fiction writers didn't just wake up one morning and decide "I'm going to write, today." No, rather they (might have) gone to college, they read good writers and they practice, practice, practice and write many drafts. This is what we must do as mathematicians.See, this is one thing that the world-at-large does not understand about real mathematics. As an example, when I told a lady at work that I was majoring in math, she said "Oh math was always my favorite subject in college." She said she liked it because there were clear steps she could "follow" to get an answer. She then said that she hated "word problems" because she wasn't any good at them. Later at lunch she said that she could do just as well in my math classes as I do if she was in the class because that we she would "learn the steps" to solve problems.Now, this particular lady is one of my favorites at work and she is really representative of how most people think of mathematics. They know that when they were in school, the were brutally forced to memorize algorithms for solving problems, so they think that upper-level mathematics is all the same but with more complex algorithms. However, this is not at all true. As you go on in math (and this is coming from a guy with 1.5 to 2 years more experience than you have so you'll learn this real fast and you have already learned some of this) you will see that mathematics becomes more like writing poetry in the sense there is no algorithm for writing poems, otherwise, we wouldn't need poets. Similarly, there are no algorithms for proving things, if there were, mathematica would have proven the Riemann Hypothesis by now.
The problems you are doing aren't "integrate so-and-so" but you will be proving things about (at times) very abstract things. I can't explain myself fully, but you will understand. But, let me leave you with some advice that I wish I had heard when I first started doing proof-based stuff. You see, I had terrible troubles understanding proofs in some of my books. I would read literally hours on perhaps one page. I thought I was an idiot for not being able to digest the stuff a lot faster. To be sure, a genius could get it much faster than I could, but I (and I am not saying this to show off) really think I am a good math student, and I realize now that it is OK to take a while to get through some proofs. You really want to understand EVERY concept before you tackle the next one.When I read a math book, I usually follow these steps:
1)Sit outside (or some other place where you can't really write anything) and read through a section of your book. Don't write anything down, and don't read any proofs or examples.
2)Do something for 20 minutes or so and try to think about why some of the lemmas/theorems you read might be true. Don't actually come up with proofs in your mind, just try to convince yourself that it is plausible that the stuff you are reading is correct.
3)Sit at a desk with your book and your paper (and, if you really want to be like a mathematician, some coffee.) Start reading the section again. When you come to a lemma/theorem, attempt to prove it yourself (don't spend too long on this if you can't come up with anything.) Then read the proof, and this time, understand every word of it before you go on. For me, it is helpful to re-
write the proof, but in my own words and usually explaining in greater detail exactly what is happening.
4)In a few days (preferably after you have done the above process for the next section or two) go over the section and try to re-construct the proofs (not from memory, but using what you have learned.) Now, doing this takes some discipline (something which I lack) and there are probably people with much more experience than I that would say this is all wrong. But, when I do it, I really feel that I have learned the material. That being said, I don't do this nearly as much as I should. I leave you with a quote by the great analyst Paul Halmos from his wonderful book: I Want To Be A Mathematician: An Automathogrophy.
"Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?"
