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.