how to write proofs

So midterms have come and gone, and now I know I need to work on mathematical exposition.

Main lesson: if you're trying to prove fact X, and facts A, B, and C imply X, then saying
"A, B, and C are true" is not a good proof (particularly if A, B, and C are out of order, and you're also assuming D, which you thought was too obvious to mention.)

Clarity is hard -- a lot harder than I gave it credit for. Bad wording can make you look confused, or worse, actually confuse you. So, some antidotes.

Leslie Lamport has an interesting article about writing structured proofs, with the large-scale points (here are my lemmas and cases) on the "coarse" level of structure, and the justifications for those points on the "fine" level.

Here's a nice article from the U. of C. about how to write proofs. It's written for beginners, but it doesn't stop being true when you're more advanced.

I had a professor once who said, "A proof is nothing more than an elaborate gesture at the truth." (I think he may have been quoting Hardy, but I can't find it anywhere.) It's just a matter of communicating what you understand explicitly enough that other people will read it, and feel they understand it too.