Rough Convergence in Normed Linear Spaces

x * ∈ X is said to be an r-limit point of a sequence (xi ) in some normed linear space (X,∥ · ∥) if (r ≥ 0). The set of all r-limit points of (xi , denoted by LIM r x i , is bounded closed and convex. Further properties, in particular the relation between this rough convergence and other convergence notions, and the dependence of LIM r x i on the roughness degree r, are investigated. For instance, the set-valued mapping r ↦ LIM r x i is strictly increasing and continuous on (), where . For a so-called ρ-Cauchy sequence (xi ) satisfying it is shown in case X = R n that r = (n/(n + 1))ρ (or for Euclidean space) is the best convergence degree such that LIM r x i ≠ Ø.