@ARTICLE{COUSIN,
    TITLE = {Cousin's Lemma},
    AUTHOR = {Coghetto, Roland},
    ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
    DAY = {31},
    MONTH = {December},
    YEAR = {2015},
    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 \cite{REAL_NS.ABS}, \cite{RSSPACE3.ABS}, \cite{LOPBAN_1.ABS} and
  version 2 the results obtained in \cite{REAL_NS.ABS},\cite{NORMSP_2.ABS}, (\textit{registrations}) \cite{LOPBAN_5.ABS})
  \par
  With the Cantor's theorem (in complete metric space) (proof by Karol P\c{a}k in \cite{COMPL_SP.ABS}), we
  formalize "The Nested Intervals Theorem in 1-dimensional Euclidean metric space"
  \par
  Pierre Cousin proof in 1892\cite{Maurey2005} the lemma, published in 1895\cite{Cousin1895}:
  "Soit, sur le plan YOX, une aire connexe S limite\'e par un contour ferme\'
  simple ou complexe; on suppose qu'\`a chaque point de S ou de son p\'erim\`etre
  correspond un cercle, de rayon non nul, ayant ce point pour centre :
  il est alors toujours possible de subdiviser S en r\´egions, en
  nombre fini et assez petites pour que chacune d'elles soit comp\`etement
  int\´erieure au cercle correspondant \`a un point convenablement choisi
  dans S ou sur son p\´erim\`etre."
  \par
  (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)\cite{Raman2015}
  \par
  The Cousin's Lemma, used in Henstock–Kurzweil integral (generalized Riemann integral), state that:
  "for any gauge $\delta$, there exists at least one $\delta$-fine tagged division".
  In the last section, we formalize this theorem.
  We use the suggestions given to the Exercice 1.I\cite[][19]{bartle2001modern} and
  with the notations: \cite{bartle1996return}\cite{yee2000integral}\cite{mawhin2001eternel}\cite{peng2008integral},
   and \cite{INTEGRA1.ABS}.},
    MSC2010 = {54D30;  03B35},
    KEYWORDS = {Cousin's lemma;Cousin's theorem;Nested Intervals Theorem}
}


@article{Maurey2005,
	abstract = {Nous nous proposons de rendre à Émile Borel le mérite d’avoir
   considéré le premier un recouvrement d’un segment de droite par une suite infinie
  d’intervalles et prouvé que l’on peut en extraire un sous-recouvrement fini. L’appellation
  de théorème de Heine-Borel souvent donnée à ce résultat, en référence à un article de Heine
  de 1872, conduit à sous-estimer les différences avec le théorème sur la continuité uniforme
  (dont une première version peut être attribuée à Dirichlet, en 1854) ; cette dénomination
  nous paraît ainsi inadéquate. En replaçant le théorème de recouvrement dans le cadre de
  la thèse où il figure, en 1894, nous rappelons qu’en l’introduisant, Borel jette en fait
  les bases d’une nouvelle théorie de la mesure.},
	author = {Maurey, Bernard, Tacchi, Jean-Pierre},
	journal = {Revue d'histoire des mathématiques},
	keywords = {Borel’s thesis; covering theorem; Hiene’s theorem; uniform continuity; measure and topology; Émile Borel; Dirichlet; Lebesgue},
	language = {fre},
	number = {2},
	pages = {163-204},
	publisher = {Société mathématique de France},
	title = {La genèse du théorème de recouvrement de Borel},
	url = {http://eudml.org/doc/252094},
	volume = {11},
	year = {2005},
}

@article{Raman2015,
     jstor_articletype = {research-article},
     title = {A Pedagogical History of Compactness},
     author = {Raman-Sundström, Manya},
     journal = {The American Mathematical Monthly},
     jstor_issuetitle = {},
     volume = {122},
     number = {7},
     jstor_formatteddate = {August–September 2015},
     pages = {pp. 619-635},
     url = {http://www.jstor.org/stable/10.4169/amer.math.monthly.122.7.619},
     ISSN = {00029890},
     abstract = {This paper traces the history of compactness from the original motivating questions
  through the development of the definition to a characterization of compactness in terms
  of nets and filters. The goal of the article is to clarify the central concepts of open-cover
  and sequential compactness, including details that a standard textbook treatment tends to leave out.},
     language = {English},
     year = {2015},
     publisher = {Mathematical Association of America},
     copyright = {Copyright © 2015 Mathematical Association of America},
    }

@article{bartle1996return,
  title={Return to the Riemann integral},
  author={Bartle, Robert G},
  journal={American Mathematical Monthly},
  pages={625--632},
  year={1996},
  publisher={JSTOR}
}

@book{bartle2001modern,
  title={A modern theory of integration},
  author={Bartle, Robert Gardner},
  volume={32},
  year={2001},
  publisher={American mathematical society Providence}
}

@book{yee2000integral,
  title={Integral: an easy approach after Kurzweil and Henstock},
  author={Yee, Lee Peng and Vyborny, Rudolf},
  volume={14},
  year={2000},
  publisher={Cambridge University Press}
}

@article{peng2008integral,
  title={The integral \`a la Henstock},
  author={Yee, Lee Peng},
  journal={Scientiae Mathematicae japonicae},
  volume={67},
  number={1},
  pages={13--21},
  year={2008}
}

@article{mawhin2001eternel,
  title={L'{\'e}ternel retour des sommes de Riemann-Stieltjes dans l'{\'e}volution du calcul int{\'e}gral},
  author={MAWHIN, Jean},
  journal={Bulletin de la Soci{\'e}t{\'e} Royale des Sciences de Li{\`e}ge},
  volume={70},
  number={4-6},
  pages={345--364},
  year={2001},
  publisher={Soci{\'e}t{\'e} royale des sciences de Li{\`e}ge}
}

@article{Cousin1895,
year={1895},
issn={0001-5962},
journal={Acta Mathematica},
volume={19},
number={1},
doi={10.1007/BF02402869},
title={Sur les fonctions den variables complexes},
url={http://dx.doi.org/10.1007/BF02402869},
publisher={Kluwer Academic Publishers},
author={Cousin, Pierre},
pages={1-61},
language={French}}


