  Let $(x_n)$ be a normalized weakly null sequence in a Banach space and let
$\varep>0$. We show that there exists a subsequence $(y_n)$ with the following
property: $$\hbox{ if }\ (a_i)\subseteq \IR\ \hbox{ and }\ F\subseteq \nat$$
satisfies $\min F\le |F|$ then $$\big\|\sum_{i\in F} a_i y_i\big\| \le
(2+\varep) \big\| \sum a_iy_i\big\|\ . $$
