The solution of the Dirichlet problem for a singularly perturbed elliptic differential equation \epsilon L_1u + L_0u = h of order 2m converges, for \epsilon \to 0 , outside of the boundary layer uniformly to a solution of the degenerate elliptic equation L_0w = h of lower order. It is shown in the case of order zero of L_0 this assertion may be proved immediately, i.e., without the usual construction of boundary layer terms, but rather elementary and on weak smoothness conditions with respect to the boundary of the domain.