At some point in every aspiring mathematician’s training and career, he/she has to “prove” something. This can be something like proving that if a function is continuous on a closed interval then it attains a maximum on that interval. Or it could be something far more complicated like proving Goldbach’s conjecture.
For the student of mathematics, the realm of mathematical rigor can be a hopeless place to be. It isn’t so much that one gets lost in that realm, it is often the case that one has no idea how to get moving — for in order to be lost, you must have at least traveled a little bit and gone off course. For many students when given a bonafide math problem or a problem that is mere play with math mechanics (here do these fifty integrals, for example) a typical reaction is “where do I start?” With math mechanics, a nudge in the “right” direction is to perhaps show a first step or ask the student to state what steps they would consider. With the math problem of the form “prove that …”, the instructor’s response often has to be carefully stated so as not to give away the proof entirely. Also, while a basic math course with its basic problems can be boiled down to a handful of steps and techniques that will likely handle every case, the same doesn’t hold true as rigidly for when having to prove something. Thus, the purpose of this write-up, is to give a set of basic guidelines for learning “how to prove” something.
I caution about the following: this write-up is not about methods of proof. For that, just google “methods of proof in mathematics” and I am sure one can find ample notes about mathematical logic (look for articles that discuss “proof by contradiction”, “proof by induction”, “direct proof”, “proof by counterexample”, “contrapositive”, “inverse”, “converse”, etc.). If you are a student of mathematics having your first taste of “proofs”, I would suggest getting a handle on how mathematical logic works.
Without further ado, in semi-logical order, here are some basic guidelines for learning how to prove something.
- Read other proofs.
- Identify steps in proofs.
- Close book.
- Reproduce other proofs one step at a time, referencing the book as needed.
- Only write what you know to be true.
- Do not try to memorize proofs, instead understand the logic of each step.
- Be able to answer “why” at any point in the proof.
- Identify conditions / assumptions in the statement of a lemma / theorem / proposition / corollary.
- Recognize the difference between an overarching assumption vs a condition from which the lemma / theorem / proposition / corollary follows. Typically an assumption is phrased as “Suppose that we have <mathematical object> …” and a condition could then be a specific constraint on that object (like continuity, boundedness, etc.).
- Relax a condition or an assumption and see where the proof breaks (in other words, where the proof required such a condition or assumption). Rinse and repeat for all conditions / assumptions.
- Do not feel that you have to prove a statement from beginning to end, in a top-down, linear fashion. There’s nothing wrong with saying, “suppose that I have statement X” and then proving something assuming you have statement X. The only thing you then have to do is go back and show that you indeed have statement X. A lot of mathematics has been developed with the desire to prove some statement X and doing so has required statements, Y, Z, A, B, C, etc. But these other statements were often not known in advance.
- Proofs worked out in textbooks are already cleaned up. Thus, that seemingly magical choice of \(\delta\) (for example) came about through an iterative process to which, unfortunately, you are not privy. It will be helpful to pretend that you don’t know specific constraints that seem like almost divine guesses — read through the proof and see where that out-of-the-hat constraint is used. Then you will most likely see why, from a readability standpoint, the constraint was stated upfront. Think of this as the writing technique known as “foreshadowing”.
- As you get more experience with proofs, then for the next statement you are to prove, try to sketch out a rough “skeletal” proof framework and fill in the gaps.
- Know your definitions and theorems exactly. “A solution” is different from “the solution”, “existence of an object” is different from actually having a specific object, etc.
- Don’t reinvent the wheel. If you have to prove X, then don’t hesitate to leverage other things that have been proven. Warning: don’t use theorem Y to prove theorem X, if the proof of theorem Y depended on theorem X (circular logic).
- Just because you are taking a Topology course, doesn’t mean you can’t use theory from elsewhere that’s relevant. Just make sure that you’re not in a circular argument. On the flip side, recognize that part of the exercise is to become comfortable with Topology, for example.
- Don’t worry about elegance at first. Just prove the damn thing.
- There’s nothing wrong with enumerating cases, but if you find yourself having to enumerate an infinite number of cases (worse yet, uncountably many cases and then, have fun enumerating), you may want to try another method (unless you are making an induction argument).
- Suppose you are given a theorem to prove. Give yourself some intuition about the statement by working out a few tests cases (by hand, even), if possible. Use sufficiently complex test cases as well as “trivial” ones.
- Corner cases will always be the most annoying and most likely to torpedo your proof. Be prepared to do some patchwork.
- Speak out loud. If possible, try to explain your proof to someone else. Just the simple act of talking out loud can help you to identify craziness in your approach.
- If you have no idea where to start, start writing definitions and theorems related to the problem statement and work out more test cases.
- It is not unusual to spend a few hours working out a proof. There are problems in mathematics that have required lifetime(s) of thought and study before they could be proven. Some current unsolved problems have been so for centuries.
- Do not measure success by quantity of theorems proved nor by speed with which theorems are proved. Volume and speed come with time and practice. The pace at which you must go is the pace at which you understand every step. There is no such thing as fast. There is no such thing as slow. There is only your pace.
If I think of more, I’ll add to this. In the meanwhile, I hope this helps! If you have some advice you’d like to share, just leave a comment!