I have just completed a first course in real analysis (studied construction of real numbers, supremum/infimum, sequences, series, limits, continuity, differentiability and very little point set topology).
My goal is to be better at analysis, to reach a point where proof-writing comes naturally to me and to fully understand and appreciate the beauty it entails.
Right now, what I want to be able to do is:
1) Polish the material I've already covered. While studying, I felt there were a lot of proofs of theorems (classical ones) which I didn't fully understand and had to take them for granted(due to limited time, I had to both, understand as well as practice so to be able to attempt quizzes, assignments and mid/final).
2) Cover further material: I want to do integration (Reiman integration, Darboux integration and Lebesgue integration) as well as move on to metric spaces and cover that as well.
My questions are:
How should I go about self-studying so the aforementioned goals are met? Should I start all over again, from scratch, or if anyone can give me a detailed plan as to studying these, that would be great!
The other question I have is related to the importance and understanding of the classical 'BIG' proofs. They are quite hard for a beginner as myself(or at least that is what I think!). I feel as if I can never approach them on my own and even understanding them is a challenge! What should I do about this? Since the ultimate goal is to become better and understand the material in a really good manner, I do want to be able to absorb the proofs for such big theorems!