  We discuss optimal constants of certain projections on subsequences of weakly
null sequences. Positive results yield constants arbitrarily close to $1$ for
Schreier type projections, and arbitrarily close to $1$ for Elton type
projections under the assumption that the weakly null sequence admits no
subsequence generating a $c_0$ spreading model. As an application, we prove
that a weakly null sequence admitting a spreading model not equivalent to the
$c_0$ basis has a quasi-greedy subsequence with quasi-greedy constant
arbitrarily close to $1$.
