The construction and convergence of an approximate solution to the initial value problem
x
′
=
f
(
t
,
x
)
,
x
(
0
)
=
x
0
x’ = f(t,x),x(0) = {x_0}
, defined on closed subsets of a Fréchet space is given. Sufficient conditions that guarantee the existence of an approximate solution are analyzed in a relation to the Nagumo boundary condition used in the Banach space case. It is indicated that the Nagumo boundary condition does not guarantee the existence of an approximate solution. Applications to fixed points are given.