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104
Topological Properties of Subsets in Real Numbers
 JOURNAL OF FORMALIZED MATHEMATICS
, 2002
"... ..."
Monotone Real Sequences. Subsequences
"... this paper. We follow the rules: n, m, k are natural numbers, r is a real number, and s 1 , s 2 , s 3 are sequences of real numbers. Let s 1 be a partial function from N to R. We say that s 1 is increasing if and only if: (Def. 1) For every n holds s 1 (n) ! s 1 (n + 1): We say that s 1 is decreasin ..."
Abstract

Cited by 99 (9 self)
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this paper. We follow the rules: n, m, k are natural numbers, r is a real number, and s 1 , s 2 , s 3 are sequences of real numbers. Let s 1 be a partial function from N to R. We say that s 1 is increasing if and only if: (Def. 1) For every n holds s 1 (n) ! s 1 (n + 1): We say that s 1 is decreasing if and only if: (Def. 2) For every n holds s 1 (n + 1) ! s 1 (n): We say that s 1 is nondecreasing if and only if: (Def. 3) For every n holds s 1 (n) s 1 (n + 1): We say that s 1 is nonincreasing if and only if: (Def. 4) For every n holds s 1 (n + 1) s 1 (n): Let f be a function. We say that f is constant if and only if: (Def. 5) For all sets n 1 , n 2 such that n 1 2 dom f and n 2 2 domf holds f(n 1 ) = f(n 2 ): Let us consider s 1 . Let us observe that s 1 is constant if and only if: (Def. 6) There exists r such that for every n holds s 1 (n) = r:
The Euclidean Space
, 1991
"... this paper. In this paper k, n are natural numbers and r is a real number. Let us consider n. The functor R ..."
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Cited by 82 (0 self)
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this paper. In this paper k, n are natural numbers and r is a real number. Let us consider n. The functor R
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 47 (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.
Real Function Continuity
, 2002
"... this paper. For simplicity, we adopt the following convention: n denotes a natural number, X , X 1 , Z, Z 1 denote sets, s, g, r, p, x 0 , x 1 , x 2 denote real numbers, s 1 denotes a sequence of real numbers, Y denotes a subset of R, and f , f 1 , f 2 denote partial functions from R to R ..."
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Cited by 46 (8 self)
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this paper. For simplicity, we adopt the following convention: n denotes a natural number, X , X 1 , Z, Z 1 denote sets, s, g, r, p, x 0 , x 1 , x 2 denote real numbers, s 1 denotes a sequence of real numbers, Y denotes a subset of R, and f , f 1 , f 2 denote partial functions from R to R
Functional Sequence from a Domain to a Domain
, 1992
"... this paper. For simplicity, we use the following convention: D, D 1 , D 2 denote non empty sets, n, k ..."
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Cited by 22 (0 self)
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this paper. For simplicity, we use the following convention: D, D 1 , D 2 denote non empty sets, n, k