We consider the limit behavior, as N --> infinity, of the expression Sigma(i = 0)(N - 1) upsilon(x(i), x(i + 1)) for programs {x(i)}(infinity)(i = 0) in a compact metric space K where upsilon is a real-valued function defined on K x K. We study the structure of (upsilon)-good programs and establish the existence of a G(delta) set F in C(K x K) such that, for each u is an element of F, all (u)-good programs have the same limit points set and also, for every x is an element of K, there exists a (u)-optimal program {x(i)}(infinity)(i = 0)satisfying x(0) = x.