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Infinitesimals via Cauchy sequences: Refining the classical equivalence
Open Mathematics, 2021 A refinement of the classic equivalence relation among Cauchy sequences yields a useful infinitesimal-enriched number system. Such an approach can be seen as formalizing Cauchy’s sentiment that a null sequence “becomes” an infinitesimal.
Bottazzi Emanuele, Katz Mikhail G.
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Metric completions, the Heine-Borel property, and approachability
Open Mathematics, 2020 We show that the metric universal cover of a plane with a puncture yields an example of a nonstandard hull properly containing the metric completion of a metric space.
Kanovei Vladimir +2 more
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Monotone subsequence via ultrapower
Open Mathematics, 2018 An ultraproduct can be a helpful organizing principle in presenting solutions of problems at many levels, as argued by Terence Tao. We apply it here to the solution of a calculus problem: every infinite sequence has a monotone infinite subsequence, and ...
Błaszczyk Piotr +3 more
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Generalized solutions of variational problems and applications
Advances in Nonlinear Analysis, 2018 Ultrafunctions are a particular class of generalized functions defined on a hyperreal field ℝ*⊃ℝ{\mathbb{R}^{*}\supset\mathbb{R}} that allow to solve variational problems with no classical solutions.
Benci Vieri +2 more
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