Complex Function Differentiability

The articles [17], [18], [3], [5], [4], [8], [2], [7], [11], [6], [16], [12], [19], [9], [10], [1], [14], [15], and [13] provide the notation and terminology for this paper. For simplicity, we use the following convention: k, n, m denote elements of N, X denotes a set, s1, s2 denote complex sequences, Y denotes a subset of C, f , f1, f2 denote partial functions from C to C, r denotes a real number, a, a1, b, x, x0, z, z0 denote complex numbers, and N1 denotes an increasing sequence of naturals. Let I be a complex sequence. We say that I is convergent to 0 if and only if: (Def. 1) I is non-zero and convergent and lim I = 0. We now state four propositions: (1) Let r1 be a sequence of real numbers and c1 be a complex sequence. If r1 = c1 and r1 is convergent, then c1 is convergent. (2) If 0 < r and for every n holds s1(n) = 1 n+r , then s1 is convergent. (3) If 0 < r and for every n holds s1(n) = 1 n+r , then lim s1 = 0. (4) If for every n holds s1(n) = 1 n+1 , then s1 is convergent and lim s1 = 0.