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How to use Zorn's lemma
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Just-do-it proofs
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Transfinite induction
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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
Lower degree by increasing dimension (or vice-versa)
Numerical optimization
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