Methods of Mathematical Proof

If the proof of a theorem is not immediately apparent, it may be because you are trying the wrong approach. Below are some effective methods of proof that may aim you in the right direction.

1. Proof by Obviousness: "The proof is so clear that it need not be mentioned."
   
2. Proof by General Agreement: "All in Favor?..."
   
3. Proof by Imagination: "Well, We'll pretend its true."
   
4. Proof by Convenience: "It would be very nice if it were true, so ..."
   
5. Proof by Necessity: "It had better be true or the whole structure of mathematics would crumble to the ground."
   
6. Proof by Plausibility: "It sounds good so it must be true."
   
7. Proof by Intimidation: "Don't be stupid, of course it's true."
   
8. Proof by Lack of Sufficient Time: "Because of the time constraint, I'll leave the proof to you."
   
9. Proof by Postponement: "The proof for this is so long and arduous, so it is given in the appendix."
   
10. Proof by Accident: "Hey, what have we here?"
    
11. Proof by Insignificance: "Who really cares anyway?"
    
12. Proof by Mumbo-Jumbo: "For any epsilon> 0 there exists a corresponding delta > 0 s.t. f(x)-L < epsilon whenever x-a < delta."
    
13. Proof by Profanity: (example omitted)
    
14. Proof by Definition: "We'll define it to be true."
    
15. Proof by Tautology: "It's true because it's true."
    
16. Proof by Plagiarism: "As we see on page 238 ..."
    
17. Proof by Lost Reference: "I know I saw this somewhere ..."
    
18. Proof by Calculus: "This proof requires calculus, so we'll skip it."
    
19. Proof by Terror: When intimidation fails ...
    
20. Proof by Lack of Interest: "Does anyone really want to see this?"
    
21. Proof by Illegibility: " ¥ ª Ð Þ þæ"
    
22. Proof by Logic: "If it is on the problem sheet, then it must be true."
    
23. Proof by Majority Rule: Only to be used if General Agreement is impossible.
    
24. Proof by Clever Variable Choice: "Let A be the number such that this proof works."
    
25. Proof by Tessellation: "This proof is just the same as the last."
    
26. Proof by Divine Word: "And the Lord said, 'Let it be true,' and it came to pass."
    
27. Proof by Stubbornness: "I don't care what you say! It is true!"
    
28. Proof by Simplification: "This proof reduces to the statement, 1 + 1 = 2."
    
29. Proof by Hasty Generalization: "Well, it works for 17, so it works for all reals."
    
30. Proof by Deception: "Now everyone turn their backs ..."
    
31. Proof by Supplication: "Oh please, let it be true."
    
32. Proof by Poor Analogy: "Well, it's just like ..."
    
33. Proof by Avoidance: Limit of Proof by Postponement as t approaches infinity.
    
34. Proof by Design: "If it's not true in today's math, invent a new system in which it is."
    
35. Proof by Intuition: "I just have this gut feeling ..."
    
36. Proof by Authority: "Well, Bill Gates says it's true, so it must be."
    
37. Proof by Vigorous Assertion: "And I REALLY MEAN THAT!"
    
38. Proof by A.F.K.T. Theorem: "Any Fool Knows That!"
    
39. Proof by vigorous handwaving: Works well in a classroom.
    
40. Proof by seduction: "Convince yourself that this is true!"
    
41. Proof by accumulated evidence: "Long and diligent search has not revealed a counterexample."
    
42. Proof by Divine Intervention: "Then a miracle occurs ..."

Source: Dick A. Wood in Mathematics Teacher (November 1998).