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Please consider this as an on-going list of techniques preferably per objective or subject.

Many mathematical books (at least lately) are focusing on "design patterns" if you like of proof-techniques or proof-schemes for similar problems/objectives.

It could be very interesting (and educational) if a comprehensive list of proof-techniques can be assembled along with some annotated comments.

Let me give some illustrative examples of what is meant:

  1. Proof of existence by actual construction of an object

  2. Proof of existence by proving impossible for the object to not exist (e.g fixed points, ergodic transformations etc..)

  3. Proof of irrationality by showing an expression is integer and at the same time between 0 and 1

  4. Proof of equality by prooving 2 opposite inequalities using known inequalities (e.g Cauchy-Swartz, Arithmetic Mean/Geometric Mean etc..)

  5. Proof of isomorphism by existence of 1-1 function between 2 structures

  6. Proof of isomorphism by counting arguments (e.g both structures have exactly same number of elements)

  7. Proofs using "Diagonalization" for impossibility results (e.g Cantor's Theorem)

  8. Proofs by generalization (a-la Grothendiek, Category theory etc.. someone can elaborate further here)

etc..

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2  
I think this is probably too broad: There are literally entire books about proof techniques. –  user61527 May 25 at 18:18
    
still i think a list of proof-techniques (even summarized) as a compendium is very very interesting (especially for proffesional mathematicians) i can also add a twist to the question if you like –  Nikos M. May 25 at 18:20
    
should i migrate this to mathoverflow, i was thinking of adding it there in the 1st place, but out of acquired speed posted it on math.se instead? –  Nikos M. May 25 at 18:26

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