The Rough Limit Set and the Core of a Real Sequence

In this paper, we prove that the ordinary core of a sequence x = (x i ) of real numbers is equal to its 2-limit set, where : = inf {r ≥ 0:LIM r x ≠ Ø}. 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.