State your game plan. A good proof begins by explaining the general line of reasoning, e.g. “We use case analysis” or “We argue by contradiction”. This creates a rough mental picture into which the reader can fit the subsequent details.
Keep a linear flow. We sometimes see proofs that are like mathematical mosaics, with juicy tidbits of reasoning sprinkled across the page. This is not good. The steps of your argument should follow one another in a sequential order.
A proof is an essay, not a calculation. Many students initially write proofs the way they compute integrals. The result is a long sequence of expressions without explantion. This is bad. A good proof usually looks like an essay with some equations thrown in. Use complete sentences.
Avoid excessive symbolism. Your reader is probably good at understanding words, but much less skilled at reading arcane mathematical symbols. So use words where you reasonably can.
Simplify. Long, complicated proofs take the reader more time and effort to understand and can more easily conceal errors. So a proof with fewer logical steps is a better proof.
Introduce notation thoughtfully. Sometimes an argument can be greatly simplified by introducing a variable, devising a special notation, or defining a new term. But do this sparingly since you’re requiring the reader to remember all that new stuff. And remember to actually define the meanings of new variables, terms, or notations; don’t just start using them!
Structure long proofs. Long programs are usually broken into a heirarchy of smaller procedures. Long proofs are much the same. Facts needed in your proof that are easily stated, but not readily proved are best pulled out and proved in preliminary lemmas. Also, if you are repeating essentially the same argument over and over, try to capture that argument in a general lemma, which you can cite repeatedly instead.
Don’t bully. Don’t use phrases like “clearly” or “obviously” in an attempt to bully the reader into accepting something which you’re having trouble proving. Also, go on the alert whenever you see one of these phrases is someone else’s proof.
Finish. At some point in a proof, you’ll have established all the essential facts you need. Resist the temptation to quit and leave the reader to draw the “obvious” conclusion. What is obvious to you as the author is not likely to be obvious to the reader. Instead, tie everything together yourself and explain why the original claim follows.
