  We prove that if $(v_i)$ is a normalized basic sequence and X is a Banach
space such that every normalized weakly null sequence in X has a subsequence
that is dominated by $(v_i)$, then there exists a uniform constant $C\geq1$
such that every normalized weakly null sequence in X has a subsequence that is
C-dominated by $(v_i)$. This extends a result of Knaust and Odell, who proved
this for the cases in which $(v_i)$ is the standard basis for $\ell_p$ or
$c_0$.
