  We provide a generalization of two results of Knaust and Odell from
\cite{KO2} and \cite{KO}. We prove that if $X$ is a Banach space and
$(g_n)_{n=1}^\infty$ is a right dominant Schauder basis such that every
normalized, weakly null sequence in $X$ admits a subsequence dominated by a
subsequence of $(g_n)_{n=1}^\infty$, then there exists a constant $C$ such that
every normalized, weakly null sequence in $X$ admits a subsequence
$C$-dominated by a subsequence of $(g_n)_{n=1}^\infty$. We also prove that if
every spreading model generated by a normalized, weakly null sequence in $X$ is
dominated by some spreading model generated by a subsequence of
$(g_n)_{n=1}^\infty$, then there exists $C$ such that every spreading model
generated by a normalized, weakly null sequence in $X$ is $C$-dominated by
every spreading model generated by a subsequence of $(g_n)_{n=1}^\infty$. We
also prove a single, ordinal-quantified result which unifies and interpolates
between these two results.
