Proof Writing and Presentation Tips

Professor: Erika L.C. King
Email: eking@hws.edu
Office: Gulick 222
Phone: (315) 781-3355

King's Home Page

Tips for discovering a good proof

1. Don't expect to be able to figure out the proof in order, from beginning to end!
2. Write down all the definitions of terms used in the statement of the problem or theorem.
3. Look for theorems that relate different properties involved in the statement. Write those down as well.
4. Come up with a concrete example to visualize or understand the concepts and make sure that you believe the statement you are trying to prove. Remember that an example is not a proof (unless you are working with an existence theorem or you are trying to disprove something), but it can help guide you to a more general argument.
5. Try to disprove the statement! Even if your goal is to prove the statement, trying to come up with a counterexample often gives you insight into the mathematical structures involved and can help lead you to a proof.
6. Write down your assumptions and what you want to show. Try working both forwards and backwards from these statements to see if you can create an argument that joins them.
7. Try direct proof first, but do not feel shy about experimenting with other types of proof like proof by contraposition or proof by contradiction.
8. Be creative! Think outside the box!

Tips for writing a good final draft of a proof

1. Begin with "Proof:" and mark the end of your proof with "Q.E.D." (which stands for "quod erat demonstrandum", Latin for "that which had to be demonstrated"), $\Box$, or some other symbol.
2. If you are doing a proof by contraposition, by contradiction, by induction or by complete induction, start with "Proof by ..." instead of just "Proof".
3. Think about what type of proof you are doing and what the two main steps are. What assumptions should you state first?
4. Avoid beginning sentences with symbols; keep symbols to a minimum in general (in written work -- they are good for presentations where they will be accompanied by spoken word).
5. Use full sentences with proper punctuation (this includes capitalizing the first word in every sentence and using periods at the ends of sentences).
6. Do not abbreviate words.
7. Justify each step.
8. Use separate paragraphs for each case/direction and make it clear which case/direction it is.
9. Define your variables before you use them. For example, say "Let $x$ be a real number greater than two." before you begin using $x$.
10. Remember that definitions are a key in connecting one idea to another. Use them. This does not mean that you need to rewrite the entire definition within your proof, just refer to it.
11. Use words carefully. Distinguish carefully between moments when you want to use "since" and moments when you want to use "if" ("since" implies that what follows IS true, whereas "if" implies that what follows is just one possible case); similarly for "and" and "implies" or "then"; and also for "the" and "a" (do you know that the noun that follows this word is the only one of its kind or is it just one of many?).
12. When you are doing an elemental argument involving unions it is often necessary to look at two separate cases, following the first one through completely before you look at the second. Be careful not to try to do too much at once.
13. Always end your proof by saying that you have proved what you set out to prove. For example, if you were trying to prove that the sum of two even numbers is even, then your last sentence of your proof should be something like: "Therefore, the sum of two even numbers is even."
14. If you are doing a proof by induction or complete induction, that means that you were able to prove what you proved by the Principle of Mathematical Induction or by the Principle of Complete Induction. Be sure that your conclusion mentions this with something like "By the Principle of Mathematical Induction,...".

Tips for giving good proof presentations

1. Read above to make sure the proof you are about to present is ready to go.
2. Begin with a verbal short summary, mentioning key theorems and definitions that will be used, before you start writing the proof on the board.
3. Write clearly and big enough for the class to see, but not too big!
4. Keep the board organized.
5. Write full sentences.
6. Pause after each sentence or two to be sure that the class is following you.
7. Avoid blocking what you are writing. This is sometimes difficult, but you can take advantage of your pauses to get out of the way.
8. Speak clearly and loud enough for everyone to hear you.
9. Your classmates are your audience. Be sure to look at them and not to present just to the professor.
10. Be genuinely enthusiastic! Enthusiasm is more important than you might imagine.