The rough limit set and the core of a real sequence


Aytar S.

NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION, vol.29, pp.283-290, 2008 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 29
  • Publication Date: 2008
  • Doi Number: 10.1080/01630560802001056
  • Title of Journal : NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION
  • Page Numbers: pp.283-290

Abstract

In this paper we prove that the ordinary core of a sequence x = (xi) of real numbers is equal to its 2 (r) over bar -Iimit set, where (r) over bar := inf{r >= 0 : LIMxr x not equal empty set}. Defining the sets r-limit inferior and r-limit superior of a sequence, we show that the r-limit set of the sequence is equal to the intersection of these sets and that r-core of the sequence is equal to the union of these sets. Finally, we prove an ordinary convergence criterion that says a sequence is convergent iff its rough core is equal to its rough limit set for the same roughness degree.