TITLE: How to stop forgetting proofs - for a first course in Real Analysis?
QUESTION [73 upvotes]: I am taking my first course in analysis. I like the subject. I study it almost on a daily basis. I try to prove theorems on my own without even looking at the hints. If I really get stuck I just read the first line of the proof and then try to continue on my own. I find this approach to be very rewarding. The problem is that I tend to forget all the proofs later. Some I still remember because I liked the idea of the proof, but most of the others I will forget fast. What am I doing wrong? Any tips on how to study analysis?

REPLY [3 votes]: I believe that if you understand the nature and behavior of real numbers then you should not have any issue with most of the proofs in real analysis. Once you know reals, you will understand what goes underneath the complicated $\epsilon-\delta$ arguments.
To understand the reals, it is necessary to have some understanding of infinity. Thus we can start with the following basic ideas:

There are an infinity of positive integers.
There is no smallest positive rational number.
A real number can be defined as a set of rationals which are less than it.
Reals have something extra compared to rationals.

There is certain technicality involved in third point above about definition of reals (how do we handle "less than real" when real itself is being defined), but this is not a major problem.
The last point about extra feature of real numbers is the most difficult to grasp and all fundamental theorems (like Bolzano-Weierstrass, Heine-Borel, Nested Interval Principle, Supremum Principle, Cauchy Completeness) refer to this one and the same extra feature in very different languages.
Most of the proofs of introductory real analysis make use of this extra property of reals. Once you understand this property very well, you should have no doubt in any of usual analysis proofs.
A very good understanding of the reals can be had by reading Dedekind's paper "Continuity and Irrational Numbers" or its exposition in first chapter of Hardy's Pure Mathematics.