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A non-trivial circular argument can often be usefully perturbed to a non-circular one
Induction front page
Use the continuity method
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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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Countable but impredicative
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I don't think this statement
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Higher dimensional analogues
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