  Let $(R,m)$ be a local Noetherian ring, let $M$ be a finitely generated
$R$-module and let $(F_{\bullet},\partial_{\bullet})$ be a free resolution of
$M$. We find a uniform bound $h$ such that the Artin-Rees containment $I^n
F_i\cap Im \, \partial_{i+1} \subseteq I^{n-h} Im \, \partial_{i+1}$ holds for
all integers $i\ge d$, for all integers $n\ge h$, and for all ideals $I$ of
$R$. In fact, we show that a considerably stronger statement holds. The uniform
bound $h$ holds for all ideals and all resolutions of $d$th syzygy modules. In
order to prove our statements, we introduce the concept of Koszul annihilating
sequences.
