  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
$n$th 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.
