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Mathematicians need to be metamathematicians
If you are getting stuck, then try to prove rigorously that your approach cannot work
What can a lower bound say about potential proofs of the upper bound?
Which techniques lead to which kinds of bounds?
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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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