# Are there any tricks to remembering proofs of mathematical theorems?

Is there a way to quickly and thoroughly remember theorems?

For example, proofs of the mean value theorem, or Rolle's theorem. Having to remember all of them off by heart has been quite tedious. Understanding helps you get the basic gist of it, but there's always some integral part of the theorem that gets left out.

• Maybe understand them first ? :) – Salem Jun 21 '16 at 18:04
• For the examples you mention, have a clear intuitive view of the geometry. – André Nicolas Jun 21 '16 at 18:15
• What @Salem says is really the answer. If you can understand what the proof actually does (what the major steps in the proof are) then you can fill in the details on your own to link them up. – MPW Jun 21 '16 at 18:15
• If I remember correctly, Grothendieck and many other mathematicians believed that any proof should be obvious and straightforward; if it seems non-trivial, one just doesn't understand what is being proved. Unfortunatelly, this doesn't always work. – lisyarus Jun 23 '16 at 17:30

Resist the urge to look them up! If you are worried you are forgetting some theorem, sit down and try to re-derive it from something you do know how to prove (maybe mvt from Rolles, which maybe you remember).

If you get stuck for awhile, look up a first step, then keep going yourself. "Actively" re-deriving theorems helps me make sure I understand them. It is easy to read something on wikipedia and say "oh ok that makes sense" without actually understanding it.

• I think one generally uses Rolle's theorem to prove MVT as a corollary, not the other way around. – J. Loreaux Jun 21 '16 at 19:50
• @J.Loreaux good call, checked rudin and he does it differently then the calc book I've been using recently, fixing. – qbert Jun 21 '16 at 20:14

Most theorems have one or two tricks and the proof just unfolds from that. I would suggest just memorizing the tricks and then just figure out how the rest comes from that.

Disbelieving the theorem is a good way to start, I find. Every time a statement is made, try to break it. Ask "what if $x=-1$?", or whatever seems likely to make an equation or an inference break down.

After all, the whole point of a proof is that the thing wasn't obvious to start with, which is why it needed proving. So put the proof to the proof.

Once you have had a good fight with the thing, you are more likely to remember how it defended itself from you. For instance, rather than trying to memorize an initial condition, you say to yourself, when one of your devious attacks fails, "Ah, that is why it insisted, at the beginning, that $a$ had to be unequal to $b$".

In the same way, believing that a given step is illogical, and then eventually understanding that it is logical after all, is more likely to burn it into your brain than trying to remember the words would.

The other technique is stretching. For instance if a theorem is about integers, will it work for rationals? Or for Gaussian integers ($a+bi$)? Stretch it and see. Occasionally a generalisation really does exist and was left out for pædagogical reasons; more often, the generalisation will fail and teach you a lot by failing.

The trick to memorizing/remembering proofs of theorems is to understand them. If you understand what a theorem is about and you remember the strategy for proving it, it is often not hard to fill in the details on the go. Some theorems do require some "tricks" that can be useful to remember.

I think the worst thing you can do is to just mindlessly memorize the words and symbols in the proof.