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36
Convergent sequences and the limit of sequences
 Journal of Formalized Mathematics
, 1989
"... Summary. The article contains definitions and same basic properties of bounded sequences (above and below), convergent sequences and the limit of sequences. In the article there are some properties of real numbers useful in the other theorems of this article. ..."
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Cited by 116 (8 self)
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Summary. The article contains definitions and same basic properties of bounded sequences (above and below), convergent sequences and the limit of sequences. In the article there are some properties of real numbers useful in the other theorems of this article.
The complex numbers
 Journal of Formalized Mathematics
, 1990
"... Summary. We define the set C of complex numbers as the set of all ordered pairs z = 〈a,b 〉 where a and b are real numbers and where addition and multiplication are defined. We define the real and imaginary parts of z and denote this by a = ℜ(z), b = ℑ(z). These definitions satisfy all the axioms for ..."
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Cited by 86 (1 self)
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Summary. We define the set C of complex numbers as the set of all ordered pairs z = 〈a,b 〉 where a and b are real numbers and where addition and multiplication are defined. We define the real and imaginary parts of z and denote this by a = ℜ(z), b = ℑ(z). These definitions satisfy all the axioms for a field. 0C = 0 + 0i and 1C = 1 + 0i are identities for addition and multiplication respectively, and there are multiplicative inverses for each non zero element in C. The difference and division of complex numbers are also defined. We do not interpret the set of all real numbers R as a subset of C. From here on we do not abandon the ordered pair notation for complex numbers. For example: i 2 = (0+1i) 2 = −1+0i � = −1. We conclude this article by introducing two operations on C which are not field operations. We define the absolute value of z denoted by z  and the conjugate of z denoted by z ∗.
Metric spaces
 Formalized Mathematics
, 1990
"... Summary. Sequences in metric spaces are defined. The article contains definitions of bounded, convergent, Cauchy sequences. The subsequences are introduced too. Some theorems concerning sequences are proved. ..."
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Cited by 45 (3 self)
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Summary. Sequences in metric spaces are defined. The article contains definitions of bounded, convergent, Cauchy sequences. The subsequences are introduced too. Some theorems concerning sequences are proved.
Bounding boxes for compact sets inE 2
 Journal of Formalized Mathematics
, 1997
"... Summary. We define pseudocompact topological spaces and prove that every compact space is pseudocompact. We also solve an exercise from [14] p.225 that for a topological space X the following are equivalent: • Every continuous real map from X is bounded (i.e. X is pseudocompact). • Every continuous ..."
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Cited by 39 (2 self)
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Summary. We define pseudocompact topological spaces and prove that every compact space is pseudocompact. We also solve an exercise from [14] p.225 that for a topological space X the following are equivalent: • Every continuous real map from X is bounded (i.e. X is pseudocompact). • Every continuous real map from X attains minimum. • Every continuous real map from X attains maximum. Finally, for a compact set in E 2 we define its bounding rectangle and introduce a collection of notions associated with the box.
Integer and rational exponents
 Formalized Mathematics
, 1991
"... Summary. The article includes definitios and theorems which are needed to define real exponent. The following notions are defined: natural exponent, integer exponent and rational exponent. ..."
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Cited by 38 (5 self)
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Summary. The article includes definitios and theorems which are needed to define real exponent. The following notions are defined: natural exponent, integer exponent and rational exponent.
The limit of a real function at infinity
 Journal of Formalized Mathematics
, 1990
"... Summary. We introduced the halflines (open and closed), real sequences divergent to infinity (plus and minus) and the proper and improper limit of a real function at infinty. We prove basic properties of halflines, sequences divergent to infinity and the limit of function at infinity. ..."
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Cited by 31 (6 self)
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Summary. We introduced the halflines (open and closed), real sequences divergent to infinity (plus and minus) and the proper and improper limit of a real function at infinty. We prove basic properties of halflines, sequences divergent to infinity and the limit of function at infinity.
The field of complex numbers
 Journal of Formalized Mathematics
"... [14], [5], [10], and [9] provide the notation and terminology for this paper. The following propositions are true: 1. PRELIMINARIES (1) For every element z of C holds z  = z. (2) For all real numbers x1, y1, x2, y2 holds (x1 + y1i) ·(x2 + y2i) = (x1 · x2 − y1 · y2)+(x1 · y2 + x2 · y1)i. (3) ..."
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Cited by 26 (1 self)
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[14], [5], [10], and [9] provide the notation and terminology for this paper. The following propositions are true: 1. PRELIMINARIES (1) For every element z of C holds z  = z. (2) For all real numbers x1, y1, x2, y2 holds (x1 + y1i) ·(x2 + y2i) = (x1 · x2 − y1 · y2)+(x1 · y2 + x2 · y1)i. (3) For every real number r holds (r+0i) · i = 0+ri. (4) For every real number r holds r+0i  = r. (5) For every element z of C such that z  � = 0 holds z+0i = z z+0i · z. 2. SOME FACTS ON THE FIELD OF COMPLEX NUMBERS Let x, y be real numbers. The functor x+yiCF yielding an element of CF is defined as follows: (Def. 1) x+yiCF = x+yi. The element iCF of CF is defined as follows: (Def. 2) iCF = i. We now state several propositions: (6) iCF = 0+1i and iCF = 0+1iCF.
Basic properties of rational numbers
 Journal of Formalized Mathematics
, 1990
"... Summary. A definition of rational numbers and some basic properties of them. Operations of addition, subtraction, multiplication are redefined for rational numbers. Functors numerator (num p) and denominator (den p) (p is rational) are defined and some properties of them are presented. Density of ra ..."
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Cited by 26 (1 self)
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Summary. A definition of rational numbers and some basic properties of them. Operations of addition, subtraction, multiplication are redefined for rational numbers. Functors numerator (num p) and denominator (den p) (p is rational) are defined and some properties of them are presented. Density of rational numbers is also given.