Skip to main content
Have a personal or library account? Click to login
Extended Real-Valued Double Sequence and Its Convergence Cover

Extended Real-Valued Double Sequence and Its Convergence

Formalized Mathematics
Volume 23 (2015): Issue 3 (September 2015)
By:   
Open Access
|Sep 2015
References

References

  1. [1] Grzegorz Bancerek. The fundamental properties of natural numbers., 1(1):41-46, 1990.
    Search in Google Scholar
  2. [2] Grzegorz Bancerek. The ordinal numbers., 1(1):91-96, 1990.
    Search in Google Scholar
  3. [3] Józef Białas. Infimum and supremum of the set of real numbers. Measure theory., 2(1):163-171, 1991.
    Search in Google Scholar
  4. [4] Józef Białas. Series of positive real numbers. Measure theory., 2(1):173-183, 1991.
    Search in Google Scholar
  5. [5] Czesław Bylinski. Binary operations., 1(1):175-180, 1990.
    Search in Google Scholar
  6. [6] Czesław Bylinski. Functions and their basic properties., 1(1): 55-65, 1990.
    Search in Google Scholar
  7. [7] Czesław Bylinski. Functions from a set to a set., 1(1):153-164, 1990.
    Search in Google Scholar
  8. [8] Czesław Bylinski. Partial functions., 1(2):357-367, 1990.
    Search in Google Scholar
  9. [9] Czesław Bylinski. Some basic properties of sets., 1(1):47-53, 1990.
    Search in Google Scholar
  10. [10] Noboru Endou. Double series and sums., 22(1):57-68, 2014. doi:10.2478/forma-2014-0006.
    Open DOISearch in Google Scholar
  11. [11] Noboru Endou and Yasunari Shidama. Integral of measurable function., 14(2):53-70, 2006. doi:10.2478/v10037-006-0008-x.
    Open DOISearch in Google Scholar
  12. [12] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Basic properties of extended real numbers., 9(3):491-494, 2001.
    Search in Google Scholar
  13. [13] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definitions and basic properties of measurable functions., 9(3):495-500, 2001.
    Search in Google Scholar
  14. [14] Noboru Endou, Keiko Narita, and Yasunari Shidama. The Lebesgue monotone convergence theorem., 16(2):167-175, 2008. doi:10.2478/v10037-008-0023-1.
    Open DOISearch in Google Scholar
  15. [15] Noboru Endou, Hiroyuki Okazaki, and Yasunari Shidama. Double sequences and limits., 21(3):163-170, 2013. doi:10.2478/forma-2013-0018.
    Open DOISearch in Google Scholar
  16. [16] Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications. Wiley, 2 edition, 1999.
    Search in Google Scholar
  17. [17] D.J.H. Garling. A Course in Mathematical Analysis: Volume 1, Foundations and Elementary Real Analysis, volume 1. Cambridge University Press, 2013.
    Open DOISearch in Google Scholar
  18. [18] Andrzej Kondracki. Basic properties of rational numbers., 1(5): 841-845, 1990.
    Search in Google Scholar
  19. [19] Jarosław Kotowicz. Monotone real sequences. Subsequences., 1 (3):471-475, 1990.
    Search in Google Scholar
  20. [20] Jarosław Kotowicz. Convergent sequences and the limit of sequences., 1(2):273-275, 1990.
    Search in Google Scholar
  21. [21] Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences., 6(2):265-268, 1997.
    Search in Google Scholar
  22. [22] Michał J. Trybulec. Integers., 1(3):501-505, 1990.
    Search in Google Scholar
  23. [23] Zinaida Trybulec. Properties of subsets., 1(1):67-71, 1990.
    Search in Google Scholar
  24. [24] Edmund Woronowicz. Relations and their basic properties., 1 (1):73-83, 1990.
    Search in Google Scholar
  25. [25] Edmund Woronowicz. Relations defined on sets., 1(1):181-186, 1990.
    Search in Google Scholar
  26. [26] Hiroshi Yamazaki, Noboru Endou, Yasunari Shidama, and Hiroyuki Okazaki. Inferior limit, superior limit and convergence of sequences of extended real numbers., 15(4):231-236, 2007. doi:10.2478/v10037-007-0026-3.
    Open DOISearch in Google Scholar
DOI: https://doi.org/10.1515/forma-2015-0021 | Journal eISSN: 1898-9934 | Journal ISSN: 1426-2630
Journal RSS Feed
Language: English
Page range: 253 - 277
Submitted on: Jul 1, 2015
Published on: Sep 30, 2015
Published by: University of Białystok
In partnership with: Paradigm Publishing Services
Keywords:
double sequence,
Fatou’s lemma for double sequence,
monotone convergence theorem for double sequence
MSC:
40A05,
40B05,
03B35
MML:
DBLSEQ 3
Related subjects:
Computer sciences,
Artificial intelligence,
Computer sciences, other,
Mathematics,
Logic and set theory,
General mathematics

© 2015 Noboru Endou, published by University of Białystok
This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License.

Volume 23 (2015): Issue 3 (September 2015)
Previous article