Convergence and Statistically Convergence in the Usual Metric Space R

Abstract

The convergence of sequences of real numbers in metric spaces is a fundamental concept widely utilized in various problem-solving and mathematical development contexts. A sequence is said to converge to a real number x if its elements approach x as they tend to infinity. In 1951, the concept of convergence was extended to include statistical convergence. A sequence is termed statistically convergent to a real number x if the proportion of its elements approaching x tends to one as the elements tend to infinity. Any sequence that converges in the usual metric space R is also statistically convergent with the same limit. Despite ongoing advancements in convergence theory, necessary conditions for the ordinary convergence of sequences in the usual metric space R have yet to be established. Consequently, this article discusses the relationship between ordinary and statistical convergence in the usual metric space R. This research explores the interplay among three convergence concepts, aiming to introduce a novel approach for determining whether a sequence converges. One of the theorems found is that if a sequence is convergent, it is also statistically convergent; however, the converse does not hold. A statistically convergent sequence will be convergent if it is monotone.