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?
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.
If you just remember individual proofs, and arrange them in a circle on your notepad, you will notice that it's $2\pi r$ to go around and remember all the proofs, but if you start at the centre (the basics) and learn the methods it's only ever $1\cdot r$ to any proof.
It's much easier if you learn the basics and the 'how to step', rather than individual proofs (and it works in all science / maths based subjects ;-)
Do not memorize proofs. Just become comfortable with how to think about certain proofs and the basic framework of proving certain ideas. That is, you do not need to memorize how to prove that $x^3$ is continuous at $a=-1$ but you do need to be familiar with how to prove continuity as a global property and prove continuity at a point.
This way of thinking helped me through proof courses. I focused on learning the process of types of proofs and how to think about proofs in general rather than learning a particular proof.
From what you have said your problem doesn't seem to be in the learning of the material, rather in the retaining of it. The solution to your situation is "Anki". Anki is a spaced repetition system application (SRS for short). Within this application you can create flashcards, to cut a long story short once you've entered these flashcards into Anki, you will rate them in difficulty, Anki will then schedule these cards so that that you do not forget them. The greater the difficult of a flashcard the greater frequency with which you'll have to review it. In this way you tend to test your memory on the specific area that you are weak on.
Anki also has a Latex function which will allow you enter mathematical expressions, etc. so it is very mathematics friendly. Personally I use it and I find that it indispensable. It helps me remember all manner of things that I would have otherwise forgotten but it does take a lot of work. You need to essentially formulate all your knowledge into Q & A format, and you need to enter it into Anki and you need to review it religiously. It requires discipline but it works.
However Anki is not a panacea, you can't just dump a load of junk into Anki and expect to remember it. You firstly need to "understand" all the material before you put it into Anki. For more information read this and download the application here.
Sometimes when working on an exercise that requires me to reference a theorem whose proof I do not remember, instead of simply citing the theorem I like to reprove the theorem using the notation in the exercise, possibly simplifying the proof when some of its parts become unnecessary.
I hope you already got a lots of good advises...one little advice from me is that some time try to understand the geometry behind those definitions like closed set connected set continuous function and all...because you already know there is a beautiful geometric structure over R as well as $ R^n$ so if you manage to relate those abstract definitions with there geometric notions then you can actually visualize all those stuffs and accordingly that you can see why those theorems are correct and writing is then just a skill to represent the geometric view in term of precise mathematical language. and of course problem solving attitude...that will help you to understand things in a more deeper way.
A last addition, do study a bit of propositional logic.
Learn the difference between
reductio ad absurdum (something is proven because its negation is not possible)
direct proof (something follows out of the givens)
proof by cases
and more that kind of constructions.
Play a bit with it and see how these proof methods reappear in your own proofs.
(reductio ad absurdum: if you set $q < 0$ and everything goes haywire then it must be $ q \ge 0$)
(direct proof if $P \to Q $ , $ Q \to R$ therefore $ P \to R $)
Do treat this lightly: just a simple book on propositional logic should be enough, don't bother at this time about predicate logic of syllogism. Only study the bare minimum. It is just to help you recognise the bones of any proof, and so that later it is easier to reproduce them.
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.
In my experience, proofs can often be summarized in one line usually describing the trick/s of the proof, by which I mean the parts of the proof which present new techniques or something.
Consider outlining proofs perhaps if they are in parts like with Lebesgue integrals.
So you may have something like
Theorem 1 Use Bolzano-Weierstrass
Prop 2 measure of null is zero
Exercise 3 Consider the union of these sets.
Theorem 4 Part I Use Simple Approximation Part II Use Monotone Convergence Part III Use Dominated Convergence
It was probably already stated but what's more important is that you understand your proofs. Remember, you're a math student not a history student. :P
How I discover a proof of a statement in analysis? I put definitions of all the terms in the statement before my eyes on a big sheet of paper, not simply relying on my own memory. Try to discover some natural consequences of the "If part". Also, try to write the "Then part" in primitive terms. First, try to progress some steps from the end side. If you stuck, then try to progress some steps from beginning side. If you stuck, then again try to progress some steps from the end side. This time, you are at a better situation, because now you can use some results of the beginning side. Again, if you stuck, then try to progress some more steps from beginning side.Experience will bold-den you Repeating this trick, the hidden proof will unfold before you slowly-slowly. A fortunate time will come when you will be able to connect both ends. Hence, the proof is complete.
Most of the above are helpful but when I took the course I had a different route. I am apparently a visual person and I visualized (and in fact wrote down on large artist's pads) every "if" condition in succession as overlapping Venn diagrams delineating truth on the inside of shapes and false on the outside. I presumed that the then part would be in the intersection of the if conditions. Or if true/false were asked decide whether the then was in the intersection. Actually writing the circles out helps keep everything in mind while narrowing the scope. Most proofs in Real Analysis consist of definitions overlapping. In fact I would take the overlap as the conclusion and prove the then was equivalent.
this is quite good method to study, when I studied the analysis, I did the exactly same thing as you are doing. I proved every theorems in the book, and read the whole book word by word many times. At last I found all my efforts worth it.
Usually the methods to prove analysis problems are related to the following topics:
- supremum and infimum principle
- monotone convergence theorem
- nested interval theorem
- Heine–Borel theorem
- Bolzano–Weierstrass theorem
- Cauchy sequence
Most of the proofs are related to one of them or some of them, they are the golden rules and the foundation of real analysis.
Try to think how every proofs are related to them and how they work together. That will be very useful.
As for some math tricky techniques, if you can come up with them, that's perfect, if not, just remember the main idea of the proofs.