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Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
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What are some examples of proving that a thing exists by proving that the set of such things...
An example from pre measure theory real analysis is in the proof of the Bounded Convergence Theorem for Riemann integrals in [1, §22.14, p288]. Here the author omits a portion of the proof, and in his …
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Nontrivial theorems with trivial proofs
Though a proof may be trivial that doesn't mean it was trivial to find it in the first place !
Euler's Rotation Theorem of elementary Euclidean Geometry (which states that for any arbitrary rigid moti …