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2015 | 23 | 3 | 253-277
Tytuł artykułu

Extended Real-Valued Double Sequence and Its Convergence

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In this article we introduce the convergence of extended realvalued double sequences [16], [17]. It is similar to our previous articles [15], [10]. In addition, we also prove Fatou’s lemma and the monotone convergence theorem for double sequences.
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autor
  • Gifu National College of Technology Gifu, Japan
Bibliografia
  • [1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.
  • [2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
  • [3] Józef Białas. Infimum and supremum of the set of real numbers. Measure theory. Formalized Mathematics, 2(1):163-171, 1991.
  • [4] Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173-183, 1991.
  • [5] Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.
  • [6] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.
  • [7] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
  • [8] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
  • [9] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
  • [10] Noboru Endou. Double series and sums. Formalized Mathematics, 22(1):57-68, 2014. doi:10.2478/forma-2014-0006.[Crossref]
  • [11] Noboru Endou and Yasunari Shidama. Integral of measurable function. Formalized Mathematics, 14(2):53-70, 2006. doi:10.2478/v10037-006-0008-x.[Crossref]
  • [12] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Basic properties of extended real numbers. Formalized Mathematics, 9(3):491-494, 2001.
  • [13] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definitions and basic properties of measurable functions. Formalized Mathematics, 9(3):495-500, 2001.
  • [14] Noboru Endou, Keiko Narita, and Yasunari Shidama. The Lebesgue monotone convergence theorem. Formalized Mathematics, 16(2):167-175, 2008. doi:10.2478/v10037-008-0023-1.[Crossref]
  • [15] Noboru Endou, Hiroyuki Okazaki, and Yasunari Shidama. Double sequences and limits. Formalized Mathematics, 21(3):163-170, 2013. doi:10.2478/forma-2013-0018.[Crossref]
  • [16] Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications. Wiley, 2 edition, 1999.
  • [17] D.J.H. Garling. A Course in Mathematical Analysis: Volume 1, Foundations and Elementary Real Analysis, volume 1. Cambridge University Press, 2013.
  • [18] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.
  • [19] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1 (3):471-475, 1990.
  • [20] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.
  • [21] Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265-268, 1997.
  • [22] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.
  • [23] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
  • [24] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.
  • [25] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
  • [26] Hiroshi Yamazaki, Noboru Endou, Yasunari Shidama, and Hiroyuki Okazaki. Inferior limit, superior limit and convergence of sequences of extended real numbers. Formalized Mathematics, 15(4):231-236, 2007. doi:10.2478/v10037-007-0026-3. [Crossref]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_forma-2015-0021
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