Integer and Rational Exponents

The terminology and notation used in this paper are introduced in the following papers: [12], [15], [4], [10], [1], [2], [3], [9], [7], [8], [14], [11], [13], [6], and [5]. For simplicity we follow the rules: a, b, c will be real numbers, m, n will be natural numbers, k, l, i will be integers, p, q will be rational numbers, and s1, s2 will be sequences of real numbers. The following propositions are true: (2)2 If s1 is convergent and for every n holds s1(n) ≥ a, then lim s1 ≥ a. (3) If s1 is convergent and for every n holds s1(n) ≤ a, then lim s1 ≤ a. Let us consider a. The functor (a)κ∈ yielding a sequence of real numbers is defined as follows: (Def.1) ((a)κ∈ )(0) = 1 and for every m holds ((a )κ∈ )(m+1) = ((a )κ∈ )(m)· a.