The purpose of this paper is to investigate the set of ε − optimal solutions to optimization problems from a metrical point of view, and generalize some results in literature that are dealing from a topological point of view. Precisely, we show that the sequence ( ) n f is epigraphical distance convergent to f if and only if for each ε > 0 , the sequence of sets (ε −arg min f n ) is epigraphical distance convergent to (ε −arg min f ) . An analogous result holds for ε − subdifferentials of convex lower semi-continuous functions defined on a Banch space and also for ε − projctions of a point to convex closed subset in X .