Convergent Real Sequences . Upper and Lower Bound of Sets of Real Numbers

The articles [9], [12], [2], [11], [4], [13], [7], [5], [1], [3], [6], [8], and [10] provide the notation and terminology for this paper. For simplicity, we adopt the following rules: n, k, m denote natural numbers, r, r1, p, g, g1, g2, s denote real numbers, 1, s2 denote sequences of real numbers, N1 denotes an increasing sequence of naturals, andX, Y denote subsets of R. Next we state several propositions: