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Proving equalities and identities
Double counting
To prove that two objects are equal show that in enough circumstances they behave in the same way
To prove that two real numbers are equal, prove that neither is greater than the other
Use basic examples to calibrate exponents
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Littlewood-Paley heuristic for derivative
Littlewood-Paley heuristic
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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