The minimal convergence and Cauchy degree of roughness via natural density

Abstract

In this article, we continue our exploration of the concepts of `minimal Cauchy degree' and `minimal convergence degree' for sequences. This investigation was first introduced in [Numer. Funct. Anal. Optim. 22 (1-2) (2001) 199-222] within the context of finite-dimensional normed spaces. However, we now take a broader approach, emphasizing the significance of natural density and infinite-dimensional normed spaces. Throughout our discussion, we extend several existing results from infinite-dimensional contexts under specific conditions. Additionally, we provide compelling examples to illustrate why some well-established results do not hold in the context of infinite-dimensional normed spaces. Finally, we examine the Jung constant $\mathcal{J}_X$ of a normed space $X$ to establish connections between statistical Cauchy degrees and statistical convergence degrees of sequences.