Double Sequences and Limits1

The notation and terminology used in this paper have been introduced in the following articles: [3], [4], [13], [5], [15], [6], [7], [16], [10], [1], [2], [8], [11], [18], [12], [17], and [9]. In this paper R, R1, R2 denote functions from N × N into R, r1, r2 denote convergent sequences of real numbers, n, m, N ,M denote natural numbers, and e, r denote real numbers. Let us consider R. We say that R is p-convergent if and only if (Def. 1) There exists a real number p such that for every real number e such that 0 < e there exists a natural number N such that for every natural numbers n, m such that n ­ N and m ­ N holds |R(n,m)− p| < e. Assume R is p-convergent. The functor P-limR yielding a real number is defined by (Def. 2) Let us consider a real number e. Suppose 0 < e. Then there exists a natural number N such that for every natural numbers n, m such that n ­ N and m ­ N holds |R(n,m)− it | < e.