Abstract

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 non-decreasing if and only if: (Def. 3) For every n holds s 1 (n) s 1 (n + 1): We say that s 1 is non-increasing 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:

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