  ``Completeness'' (i.e. probability conservation) is not usually satisfied in
the cumulant expansion of the Anderson lattice when a reduced state space is
employed for $U\to \infty $. To understand this result, the well known
``Chain'' approximation is first calculated for finite $U$, followed by taking
$U\to \infty $. Completeness is recovered by this procedure, but this result
hides a serious inconsistency that causes completeness failure in the reduced
space calculation. Completeness is satisfied and the inconsistency is removed
by choosing an adequate family of diagrams. The main result of this work is
that using a reduced space of relevant states is as good as using the whole
space.
