# Inferior Limit, Superior Limit and Convergence of Sequences of Extended Real Numbers

Hiroshi Yamazaki; Noboru Endou; Yasunari Shidama; Hiroyuki Okazaki

Formalized Mathematics (2007)

- Volume: 15, Issue: 4, page 231-235
- ISSN: 1426-2630

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topHiroshi Yamazaki, et al. "Inferior Limit, Superior Limit and Convergence of Sequences of Extended Real Numbers." Formalized Mathematics 15.4 (2007): 231-235. <http://eudml.org/doc/267112>.

@article{HiroshiYamazaki2007,

abstract = {In this article, we extended properties of sequences of real numbers to sequences of extended real numbers. We also introduced basic properties of the inferior limit, superior limit and convergence of sequences of extended real numbers.},

author = {Hiroshi Yamazaki, Noboru Endou, Yasunari Shidama, Hiroyuki Okazaki},

journal = {Formalized Mathematics},

language = {eng},

number = {4},

pages = {231-235},

title = {Inferior Limit, Superior Limit and Convergence of Sequences of Extended Real Numbers},

url = {http://eudml.org/doc/267112},

volume = {15},

year = {2007},

}

TY - JOUR

AU - Hiroshi Yamazaki

AU - Noboru Endou

AU - Yasunari Shidama

AU - Hiroyuki Okazaki

TI - Inferior Limit, Superior Limit and Convergence of Sequences of Extended Real Numbers

JO - Formalized Mathematics

PY - 2007

VL - 15

IS - 4

SP - 231

EP - 235

AB - In this article, we extended properties of sequences of real numbers to sequences of extended real numbers. We also introduced basic properties of the inferior limit, superior limit and convergence of sequences of extended real numbers.

LA - eng

UR - http://eudml.org/doc/267112

ER -

## References

top- [4] Józef Białas. The σ-additive measure theory. Formalized Mathematics, 2(2):263-270, 1991.
- [5] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
- [6] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
- [7] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
- [8] Czesław Byliński and Piotr Rudnicki. Bounding boxes for compact sets in ε2. Formalized Mathematics, 6(3):427-440, 1997.
- [9] Noboru Endou and Yasunari Shidama. Integral of measurable function. Formalized Mathematics, 14(2):53-70, 2006.
- [10] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definitions and basic properties of measurable functions. Formalized Mathematics, 9(3):495-500, 2001.
- [11] Adam Grabowski. On the Kuratowski limit operators. Formalized Mathematics, 11(4):399-409, 2003.
- [12] Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477-481, 1990.
- [13] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.
- [14] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3):471-475, 1990.
- [15] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.
- [16] Andrzej Nedzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401-407, 1990.
- [17] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.
- [18] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9-11, 1990.
- [19] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
- [20] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.
- [21] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
- [22] Bo Zhang, Hiroshi Yamazaki, and Yatsuka Nakamura. Inferior limit and superior limit of sequences of real numbers. Formalized Mathematics, 13(3):375-381, 2005.
- [1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
- [2] Józef Białas. Infimum and supremum of the set of real numbers. Measure theory. Formalized Mathematics, 2(1):163-171, 1991.
- [3] Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173-183, 1991.

## Citations in EuDML Documents

top- Noboru Endou, Keiko Narita, Yasunari Shidama, Fatou's Lemma and the Lebesgue's Convergence Theorem
- Noboru Endou, Hiroyuki Okazaki, Yasunari Shidama, Hopf Extension Theorem of Measure
- Noboru Endou, Yasunari Shidama, Keiko Narita, Egoroff's Theorem
- Noboru Endou, Keiko Narita, Yasunari Shidama, The Lebesgue Monotone Convergence Theorem
- Keiko Narita, Noboru Endou, Yasunari Shidama, The Measurability of Complex-Valued Functional Sequences
- Noboru Endou, Extended Real-Valued Double Sequence and Its Convergence
- Noboru Endou, Product Pre-Measure

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