Double Sequences and Limits

Abstract

Summary Double sequences are important extension of the ordinary notion of a sequence. In this article we formalized three types of limits of double sequences and the theory of these limits.

Double Sequences and Iterated Limits in Regular Space

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First, we define in Mizar [5], the Cartesian product of two filters bases and the Cartesian product of two filters. After comparing the product of two Fréchet filters on N (F1) with the Fréchet filter on N × N (F2), we compare limF 1 and limF 2 for all double sequences in a non empty topological space.Endou, Okazaki and Shidama formalized in [14] the "convergence in Pringsheim's sense" for double sequence of real numbers. We show some basic correspondences between the p-convergence and the filter convergence in a topological space.

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“…We show some basic correspondences between the p-convergence and the filter convergence in a topological space. Then we formalize that the double sequence (xm,n = 1 m+1 ) (m,n) ∈ N × N converges in "Pringsheim's sense" but not in Frechet filter on N × N sense.In the next section, we generalize some definitions: "is convergent in the first coordinate", "is convergent in the second coordinate", "the lim in the first coordinate of", "the lim in the second coordinate of" according to [14], in Hausdorff space.Finally, we generalize two theorems: (3) and ( 4) from [14] in the case of double sequences and we formalize the "iterated limit" theorem ("Double limit" [7], p. 81, par. 8.5 "Double limite" [6] (TG I,57)), all in regular space.…”

“…Endou, Okazaki and Shidama formalized in [14] the "convergence in Pringsheim's sense" for double sequence of real numbers. We show some basic correspondences between the p-convergence and the filter convergence in a topological space.…”

“…In the next section, we generalize some definitions: "is convergent in the first coordinate", "is convergent in the second coordinate", "the lim in the first coordinate of", "the lim in the second coordinate of" according to [14], in Hausdorff space.…”

Extended Real-Valued Double Sequence and Its Convergence

“…The notation and terminology used in this paper have been introduced in the following articles: [5], [21], [15], [10], [12], [6], [7], [22], [13], [11], [14], [1], [2], [8], [18], [24], [25], [26], [20], [23], [3], [4], and [9].…”