The rough limit set and the core of a real sequence


Aytar S.

NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION, cilt.29, ss.283-290, 2008 (SCI İndekslerine Giren Dergi) identifier identifier

  • Cilt numarası: 29
  • Basım Tarihi: 2008
  • Doi Numarası: 10.1080/01630560802001056
  • Dergi Adı: NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION
  • Sayfa Sayısı: ss.283-290

Özet

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.