  We give a constructive proof of the classical Cauchy-Kovalevskaya theorem in
the ODE setting which provides a sufficient condition for an initial value
problem to have a unique analytic solution. Our proof is inspired by a modern
functional analytic technique for rigorously solving nonlinear problems known
as the radii polynomial approach. The main idea is to recast the existence and
uniqueness of analytic solutions as a fixed point problem for an appropriately
chosen Banach space. A key aspect of this method is the usage of an approximate
solution which plays a crucial role in the theoretical proof. Our proof is
constructive in the sense that we provide an explicit recipe for constructing
the fixed point problem, an approximate solution, and the bounds necessary to
prove the existence of the fixed point.
