How can I improve my abstract thinking in real analysis?

[geoman]

Im currently taking an introduction to real analysis class and here's the problem. I do very well with all the math I've encountered before this. I'm really not doing well with the abstract nature of real analysis. I'm having trouble proving things in general because of the fact that i have choices in the way i prove them. I guess I am just not a very abstract thinker and I'm wondering if anyone has had the same problem and has a suggestion as to how i can study differently or w/e

thanks

 

[phreak]

The only way to study analysis is to get an intuitive grasp of it. This was supposed to be what calculus was all about too, but then that went to hell and calculus became about evaluating integrals instead of actually learning the concepts, which are really the only thing you need in calculus (since we have TI-89s to do our bidding).

Everything in analysis is very geometric, even though it's taught in rigorous mathematical language. Once you understand the geometric notions, there's really no problem: to solve problems, just solve them geometrically, then convert your thoughts into mathematics. This takes a lot of time to get used to, but after a few months, you'll think it's trivial.

 

[owlpride]

I agree with phreak. In an elementary analysis course, the best way to approach most proofs is to draw a picture or two. Maybe try constructing a counterexample and see why you cannot. Or omit the hypotheses one by one and construct a counterexample for each case, to see why you need every single hypothesis. Once you have an intuitive idea of why a theorem is true, try to convert your thoughts into a rigorous argument. Luckily that's a lot more straight forward in elementary analysis than in a lot of other branches of math.