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Proving "for all" statements
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A non-trivial circular argument can often be usefully perturbed to a non-circular one
I have a problem about a continuous function
LU factorization
nlogn-type bounds
Prove the result for some cases and deduce it for the rest
Proving "for all" statements
Use basic examples to calibrate exponents
Use the continuity method
Use the implicit function theorem to prove smoothness
Using generators and closure properties
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Littlewood-Paley heuristic for derivative
Geometric view of Hölder's inequality
Diagonal arguments
Finding an interval for rational numbers with a high denominator
Try to prove a stronger result
Use self-similarity to get a limit from an inferior or superior limit.
Prove a consequence first
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