Hilbert space frames containing a Riesz basis and Banach spaces which have no subspace isomorphic to c0c_0

We prove that a Hilbert space frame \fti contains a Riesz basis if every subfamily \ftj , J \subseteq I , is a frame for its closed span. Secondly we give a new characterization of Banach spaces which do not have any subspace isomorphic to $c_0$. This result immediately leads to an improvement of a recent theorem of Holub concerning frames consisting of a Riesz basis plus finitely many elements