Аннотация:We construct a sequence that converges to a solution of the Cauchy problem for a singularly perturbed linear inhomogeneous differential equation of an arbitrary order. This sequence is also an asymptotic sequence in the following sense: the deviation (in the norm of the space of continuous functions) of its nth element from the solution of the problem is proportional to the (n + 1)th power of the parameter of perturbation. This sequence can be used for justification of asymptotics obtained by the method of boundary functions.