## Convergent Real Sequences. Upper and Lower Bound of Sets of Real Numbers (2000)

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Citations: | 87 - 4 self |

### BibTeX

@MISC{Kotowicz00convergentreal,

author = {Jarosław Kotowicz},

title = {Convergent Real Sequences. Upper and Lower Bound of Sets of Real Numbers},

year = {2000}

}

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### Abstract

### Citations

1290 | Tarski Grothendieck set theory
- Trybulec
- 1990
(Show Context)
Citation Context ...m, Cauchy theorem and others. Bounded sets of real numbers and lower and upper bound of subset of real numbers are defined. MML Identifier:SEQ_4. WWW:http://mizar.org/JFM/Vol1/seq_4.html The articles =-=[9]-=-, [12], [2], [11], [4], [13], [7], [5], [1], [3], [6], [8], and [10] provide the notation and terminology for this paper. For simplicity, we adopt the following rules: n, k, m denote natural numbers, ... |

1222 | Properties of subsets
- Trybulec
- 1990
(Show Context)
Citation Context ...uchy theorem and others. Bounded sets of real numbers and lower and upper bound of subset of real numbers are defined. MML Identifier:SEQ_4. WWW:http://mizar.org/JFM/Vol1/seq_4.html The articles [9], =-=[12]-=-, [2], [11], [4], [13], [7], [5], [1], [3], [6], [8], and [10] provide the notation and terminology for this paper. For simplicity, we adopt the following rules: n, k, m denote natural numbers, r, r1,... |

1044 | Functions from a set to a set
- Byliński
- 1990
(Show Context)
Citation Context ...al numbers and lower and upper bound of subset of real numbers are defined. MML Identifier:SEQ_4. WWW:http://mizar.org/JFM/Vol1/seq_4.html The articles [9], [12], [2], [11], [4], [13], [7], [5], [1], =-=[3]-=-, [6], [8], and [10] provide the notation and terminology for this paper. For simplicity, we adopt the following rules: n, k, m denote natural numbers, r, r1, p, g, g1, g2, s denote real numbers, s1, ... |

1019 | Relations and their basic properties
- Woronowicz
- 1990
(Show Context)
Citation Context ...s. Bounded sets of real numbers and lower and upper bound of subset of real numbers are defined. MML Identifier:SEQ_4. WWW:http://mizar.org/JFM/Vol1/seq_4.html The articles [9], [12], [2], [11], [4], =-=[13]-=-, [7], [5], [1], [3], [6], [8], and [10] provide the notation and terminology for this paper. For simplicity, we adopt the following rules: n, k, m denote natural numbers, r, r1, p, g, g1, g2, s denot... |

666 | The ordinal numbers
- Bancerek
- 1990
(Show Context)
Citation Context ...heorem and others. Bounded sets of real numbers and lower and upper bound of subset of real numbers are defined. MML Identifier:SEQ_4. WWW:http://mizar.org/JFM/Vol1/seq_4.html The articles [9], [12], =-=[2]-=-, [11], [4], [13], [7], [5], [1], [3], [6], [8], and [10] provide the notation and terminology for this paper. For simplicity, we adopt the following rules: n, k, m denote natural numbers, r, r1, p, g... |

660 | The fundamental properties of natural numbers
- Bancerek
- 1990
(Show Context)
Citation Context ...of real numbers and lower and upper bound of subset of real numbers are defined. MML Identifier:SEQ_4. WWW:http://mizar.org/JFM/Vol1/seq_4.html The articles [9], [12], [2], [11], [4], [13], [7], [5], =-=[1]-=-, [3], [6], [8], and [10] provide the notation and terminology for this paper. For simplicity, we adopt the following rules: n, k, m denote natural numbers, r, r1, p, g, g1, g2, s denote real numbers,... |

559 | Basic properties of real numbers
- Hryniewiecki
- 1990
(Show Context)
Citation Context ...others. Bounded sets of real numbers and lower and upper bound of subset of real numbers are defined. MML Identifier:SEQ_4. WWW:http://mizar.org/JFM/Vol1/seq_4.html The articles [9], [12], [2], [11], =-=[4]-=-, [13], [7], [5], [1], [3], [6], [8], and [10] provide the notation and terminology for this paper. For simplicity, we adopt the following rules: n, k, m denote natural numbers, r, r1, p, g, g1, g2, s... |

370 |
Subsets of real numbers
- Trybulec
(Show Context)
Citation Context ...m and others. Bounded sets of real numbers and lower and upper bound of subset of real numbers are defined. MML Identifier:SEQ_4. WWW:http://mizar.org/JFM/Vol1/seq_4.html The articles [9], [12], [2], =-=[11]-=-, [4], [13], [7], [5], [1], [3], [6], [8], and [10] provide the notation and terminology for this paper. For simplicity, we adopt the following rules: n, k, m denote natural numbers, r, r1, p, g, g1, ... |

229 |
sequences and basic operations on them
- Real
- 1990
(Show Context)
Citation Context ...nded sets of real numbers and lower and upper bound of subset of real numbers are defined. MML Identifier:SEQ_4. WWW:http://mizar.org/JFM/Vol1/seq_4.html The articles [9], [12], [2], [11], [4], [13], =-=[7]-=-, [5], [1], [3], [6], [8], and [10] provide the notation and terminology for this paper. For simplicity, we adopt the following rules: n, k, m denote natural numbers, r, r1, p, g, g1, g2, s denote rea... |

120 | Convergent sequences and the limit of sequences
- Kotowicz
- 1990
(Show Context)
Citation Context .... Upper and Lower Bound of Sets of Real Numbers Jarosław Kotowicz Warsaw University Białystok Summary. The article contains theorems about convergent sequences and the limit of sequences occurring in =-=[5]-=- such as Bolzano-Weierstrass theorem, Cauchy theorem and others. Bounded sets of real numbers and lower and upper bound of subset of real numbers are defined. MML Identifier:SEQ_4. WWW:http://mizar.or... |

104 | Some properties of functions modul and signum - Popio̷lek - 1990 |

99 | Monotone real sequences. Subsequences
- Kotowicz
- 1990
(Show Context)
Citation Context ...mbers and lower and upper bound of subset of real numbers are defined. MML Identifier:SEQ_4. WWW:http://mizar.org/JFM/Vol1/seq_4.html The articles [9], [12], [2], [11], [4], [13], [7], [5], [1], [3], =-=[6]-=-, [8], and [10] provide the notation and terminology for this paper. For simplicity, we adopt the following rules: n, k, m denote natural numbers, r, r1, p, g, g1, g2, s denote real numbers, s1, s2 de... |

89 |
Some properties of functions modul and signum
- Popiołek
- 1989
(Show Context)
Citation Context ... and lower and upper bound of subset of real numbers are defined. MML Identifier:SEQ_4. WWW:http://mizar.org/JFM/Vol1/seq_4.html The articles [9], [12], [2], [11], [4], [13], [7], [5], [1], [3], [6], =-=[8]-=-, and [10] provide the notation and terminology for this paper. For simplicity, we adopt the following rules: n, k, m denote natural numbers, r, r1, p, g, g1, g2, s denote real numbers, s1, s2 denote ... |

42 |
On the sets inhabited by numbers
- Trybulec
(Show Context)
Citation Context ...r and upper bound of subset of real numbers are defined. MML Identifier:SEQ_4. WWW:http://mizar.org/JFM/Vol1/seq_4.html The articles [9], [12], [2], [11], [4], [13], [7], [5], [1], [3], [6], [8], and =-=[10]-=- provide the notation and terminology for this paper. For simplicity, we adopt the following rules: n, k, m denote natural numbers, r, r1, p, g, g1, g2, s denote real numbers, s1, s2 denote sequences ... |