Cousin’s Lemma

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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].

[1] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.

[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[4] Robert G. Bartle. Return to the Riemann integral. American Mathematical Monthly, pages 625–632, 1996.

[5] Robert G. Bartle. A modern theory of integration, volume 32. American Mathematical Society Providence, 2001.

[6] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.

[7] Czesław Byliński. Some properties of restrictions of finite sequences. Formalized Mathematics, 5(2):241–245, 1996.

[8] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.

[9] Pierre Cousin. Sur les fonctions de n variables complexes. Acta Mathematica, 19(1):1–61, 1895. doi:10.1007/BF02402869.

[10] Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599–603, 1991.

[11] Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces – fundamental concepts. Formalized Mathematics, 2(4):605–608, 1991.

[12] Noboru Endou and Artur Korniłowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics, 8(1):93–102, 1999.

[13] Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577–580, 2005.

[14] Noboru Endou, Yasunari Shidama, and Katsumasa Okamura. Baire’s category theorem and some spaces generated from real normed space. Formalized Mathematics, 14(4): 213–219, 2006. doi:10.2478/v10037-006-0024-x.

[15] Adam Grabowski and Yatsuka Nakamura. Some properties of real maps. Formalized Mathematics, 6(4):455–459, 1997.

[16] Artur Korniłowicz. Properties of connected subsets of the real line. Formalized Mathematics, 13(2):315–323, 2005.

[17] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.

[18] Bernard Maurey and Jean-Pierre Tacchi. La genèse du théorème de recouvrement de Borel. Revue d’histoire des mathématiques, 11(2):163–204, 2005.

[19] Jean Mawhin. L’éternel retour des sommes de Riemann-Stieltjes dans l’évolution du calcul intégral. Bulletin de la Société Royale des Sciences de Liège, 70(4–6):345–364, 2001.

[20] Yatsuka Nakamura and Andrzej Trybulec. A decomposition of a simple closed curves and the order of their points. Formalized Mathematics, 6(4):563–572, 1997.

[21] Robin Nittka. Conway’s games and some of their basic properties. Formalized Mathematics, 19(2):73–81, 2011. doi:10.2478/v10037-011-0013-6.

[22] Karol Pąk. Complete spaces. Formalized Mathematics, 16(1):35–43, 2008. doi:10.2478/v10037-008-0006-2.

[23] Manya Raman-Sundström. A pedagogical history of compactness. The American Mathematical Monthly, 122(7):619–635, 2015.

[24] Hideki Sakurai, Hisayoshi Kunimune, and Yasunari Shidama. Uniform boundedness principle. Formalized Mathematics, 16(1):19–21, 2008. doi:10.2478/v10037-008-0003-5.

[25] Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39–48, 2004.

[26] Yasumasa Suzuki, Noboru Endou, and Yasunari Shidama. Banach space of absolute summable real sequences. Formalized Mathematics, 11(4):377–380, 2003.

[27] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.

[28] Lee Peng Yee. The integral à la Henstock. Scientiae Mathematicae Japonicae, 67(1): 13–21, 2008.

[29] Lee Peng Yee and Rudolf Vyborny. Integral: an easy approach after Kurzweil and Henstock, volume 14. Cambridge University Press, 2000.

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