Part 1 should start as: let . Then for some ... Part 2 should start as: let . Now, by surjectivity of , there exists an such that ...
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1) Let . Then for some . , hence if then . 2) Let . Now, by surjectivity of , there exists an such that . Since and we have that and so . Not too pleased with the last one.. Thanks!
Originally Posted by ojones Plato: "Proofs by contradiction should not be used if a direct proof is available" - Paul Halmos on mathematical writing. "All students are enjoined in the strongest possible terms to eschew proofs by contradiction!" - Halsey Royden in Real Analysis. By the mathematicians we picked I guess it's Topology vs. Measure Theory :P
Not really Drexel28. I was just trying to find comments by known mathematicians on the subject. RL Moore is more famous for teaching than research and his comments should be taken in that context.
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