how to be good at proving?

I do not consider myself "good" at proving things. However, I know that I have gotten better. The key to writing a proof is understanding what you are trying to prove, which is harder than it may seem.

Know your definitions. Often, I have been hampered or seen students hampered by not really knowing all of the definitions in the problem statement.

Work with others. Look at what someone else has done in a proof and ask questions. Ask how they came up with the idea, ask that person to explain the proof to you. Also, do the same for them. Explain your proofs to a classmate and have them ask you questions.

Try everything. Students often get stuck on proofs because they try one idea that does not work and give up. I often go through several bad ideas before getting anywhere on a proof. Another good strategy is to work with specific examples until you understand the problem. Plug in numbers and see why the theorem seems to be true. Also, try to construct a counterexample. The reason counterexamples fail often leads to a way to prove the statement.


Practice, Practice, Practice!

Get books in the class you are doing, review the proofs. Learn to look at a theorem and see if you can figure out a proof approach.

There are also books that may help along these lines with general proof approaches.

  • General Proof Strategies How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library) [Paperback] G. Polya (Author)

  • How to Prove It: A Structured Approach [Paperback] Daniel J. Velleman (Author)

  • The Nuts and Bolts of Proofs, Third Edition: An Introduction to Mathematical Proofs [Paperback] Antonella Cupillari (Author)

  • How to Read and Do Proofs: An Introduction to Mathematical Thought Processes [Paperback] Daniel Solow (Author)

  • Discrete Math http://www.cs.dartmouth.edu/~ac/Teach/CS19-Winter06/SlidesAndNotes/lec12induction.pdf Discrete Mathematics with Proof [Hardcover] Eric Gossett (Author)

  • Discrete Mathematics: Mathematical Reasoning and Proof with Puzzles, Patterns, and Games [Hardcover] Douglas E. Ensley (Author), J. Winston Crawley (Author)

  • Schaum's Outline of Discrete Mathematics, Revised Third Edition (Schaum's Outline Series) by Seymour Lipschutz and Marc Lipson (Aug 26, 2009)

  • 2000 Solved Problems in Discrete Mathematics by Seymour Lipschutz (Oct 1, 1991)

  • Concrete Mathematics: A Foundation for Computer Science (2nd Edition) by Ronald L. Graham, Donald E. Knuth and Oren Patashnik (Mar 10, 1994)

  • Finite and Discrete Math Problem Solver (REA) (Problem Solvers Solution Guides) by The Editors of REA and Lutfi A. Lutfiyya (Jan 25, 1985)

The problem books above would also be useful references for working problems and proofs.

HTH ~A


While you mention proof methods, what you seem to need are proof-finding strategies. That's a large field. Here are just a few hints:

  • Make yourself acquated with the premises. How can the statement fail if a single premise is left out?
  • Find yourself a specific numerical example of the problem statement and check the conditions. Maybe you note a way how the premises enforce the validity of the statement for this example.
  • Try to find a counterexample. You (probably) won't find one, but you might notice what kind of obstacles prevent you from finding it.
  • Check extremes. If the statement says "For all real numbers with $0<r<2$ ...", then check what would happen with $r=0$ and $r=2$

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Proof Writing

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