  We consider the cubic and quintic Gross-Pitaevskii (GP) hierarchies in
$d\geq1$ dimensions, for focusing and defocusing interactions. We present a new
proof of existence of solutions that does not require the a priori bound on the
spacetime norm, which was introduced in the work of Klainerman and Machedon,
\cite{klma}, and used in our earlier work, \cite{chpa2}.
