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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. 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