It remains to prove that
with
implies (c). The statement
is trivial. We need to prove
.
Let
so
that
We seek
such
that
or

Thus we need to have
and
under assumptions that
.
Furthermore, by the condition (2), the summation in
and the number of
is finite. Hence,
is finite dimensional linear equation. The matrix of
has the
form
for some index ranges
and
of the same length depending on diameter of
.
Since
has only a finite number non-zero coordinates, the matrix
is semi-diagonal. Hence,
and
always have a solution.