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Formalized Mathematics

(a computer assisted approach)

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Volume 24, Issue 2 (Jun 2016)

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Cousin’s Lemma

Roland Coghetto
  • Rue de la Brasserie 5, 7100 La Louvière, Belgium
Published Online: 2016-12-08 | DOI: https://doi.org/10.1515/forma-2016-0009

Summary

We formalize, in two different ways, that “the n-dimensional Euclidean metric space is a complete metric space” (version 1. with the results obtained in [13], [26], [25] and version 2., the results obtained in [13], [14], (registrations) [24]).

With the Cantor’s theorem - in complete metric space (proof by Karol Pąk in [22]), we formalize “The Nested Intervals Theorem in 1-dimensional Euclidean metric space”.

Pierre Cousin’s proof in 1892 [18] the lemma, published in 1895 [9] states that:

“Soit, sur le plan YOX, une aire connexe S limitée par un contour fermé simple ou complexe; on suppose qu’à chaque point de S ou de son périmètre correspond un cercle, de rayon non nul, ayant ce point pour centre : il est alors toujours possible de subdiviser S en régions, en nombre fini et assez petites pour que chacune d’elles soit complétement intérieure au cercle correspondant à un point convenablement choisi dans S ou sur son périmètre.”

(In the plane YOX let S be a connected area bounded by a closed contour, simple or complex; one supposes that at each point of S or its perimeter there is a circle, of non-zero radius, having this point as its centre; it is then always possible to subdivide S into regions, finite in number and sufficiently small for each one of them to be entirely inside a circle corresponding to a suitably chosen point in S or on its perimeter) [23].

Cousin’s Lemma, used in Henstock and Kurzweil integral [29] (generalized Riemann integral), state that: “for any gauge δ, there exists at least one δ-fine tagged partition”. In the last section, we formalize this theorem. We use the suggestions given to the Cousin’s Theorem p.11 in [5] and with notations: [4], [29], [19], [28] and [12].

Keywords: Cousin’s lemma; Cousin’s theorem; nested intervals theorem

MSC 2010: 54D30; 03B35

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About the article

Received: 2015-12-31

Published Online: 2016-12-08

Published in Print: 2016-06-01


Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, ISSN (Print) 1426-2630, DOI: https://doi.org/10.1515/forma-2016-0009.

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© 2016 Roland Coghetto, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. BY-SA 3.0

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