How to stop forgetting proofs - for a first course in Real Analysis?

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?

Solution 1:

Don't try to memorise the proofs: try to memorise the methods that are used in most analysis proofs. That way you only have to memorise a handful of methods instead of 30-50 proofs, and you can adapt them to prove things you have never seen before as well.

Solution 2:

One thing that works for me when learning a theorem is to go through all the conditions and find corresponding counter-examples, as well as seeing exactly where the proof fails. Take for instance Rolle's theorem:

If a real-valued function $f$ is continuous on a closed interval $[a, b]$, differentiable on the open interval $(a, b)$, and $f(a) = f(b)$, then there exists at least one $c$ in the open interval $(a, b)$ such that $$f'(c) = 0.$$

Find a function which is continuous on $[a,b]$, but which is not differentiable on $(a,b)$, and for which the statement does not hold. Work through the proof using this counter-example, and spot the bit when the proof fails. Then find a function which is differentiable on $(a,b)$ but which is not continuous on $[a,b]$, and do the same.

Make sure you can find examples which are as sharp as possible - like making sure that you can find a function which is differentiable everywhere on $(a,b)$ but a single point. Make sure as well that they are as general as possible - given any point in $(a,b)$, can you find a function which is not differentiable at that point, and for which the statement of the theorem is not true? You will start going from specific, concrete counterexamples, to 'schemas' or ways of generating families, hopefully exhaustive families of counterexamples. (This is often how counterexamples are expressed in more advanced mathematical literature.)

You should do this for every condition in the theorem. The counterexample where $f(a) \neq f(b)$ might seem obvious, but you still need to understand exactly where the theorem breaks.

When you study several dependent theorems and lemmas, you will start to see how they interlock through these conditions and counterexamples. So the Mean Value Theorem has the same conditions as Rolle's theorem, because you use Rolle's theorem to prove the MVT. But the function to which we apply Rolle's theorem is not the same function that we are proving the MVT for, so how can we be sure that the same conditions hold for both? There are lots of pairs of theorems where one has a strictly stronger condition and a strictly stronger statement - you can compare these to see the tradeoff between the two.

When you do calculus; functions, intervals, variables are the objects which you are learning about and which you learn to manipulate in exercises until you become familiar with how they work. In analysis proofs; lemmas, theorems, conditions and statements are the things you work with. You need to learn to manipulate and understand these rather than just form a mental picture of functions, series, etc. - the things they are ostensibly 'about'.

You might also find that many of the counterexamples to analysis theorems are very interesting objects in their own right, for example the middle thirds Cantor set, the Cantor staircase, the indicator function of the rationals, etc.