Rough convergence in normed linear spaces

x(*) is an element of X is Said to be an r-limit point of a sequence (x(i)) in some normed linear space (X, //.//) if For All epsilon > 0 There Existsi(epsilon) is an element of N : i greater than or equal to i epsilon double right arrow \\x(i) - x(*)\\ less than or equal to r + epsilon (r greater than or equal to 0). The set of all r-limit points of (x(i)), 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 ((r) over bar, + infinity), where (r) over bar := inf{r is an element of R+: LIM(r)x(i) not equal 0}. For a so-called p-Cauchy sequence (x(i)) satisfying For All epsilon > 0 There Existsi(epsilon) : i,j greater than or equal to i(epsilon) double right arrow \\x(i) - x(j)\\ < rho + epsilon, it is shown in case X = R-n that r = (n/(n + 1))rho (or r = rootn/2(n + 1)rho for Euclidean space) is the best convergence degree such that LIM(r)x(i) not equal 0.