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Using a tight-binding model that takes into account a realistic electronic band structure and includes defect scattering we investigate spin-dependent transport in Co/Cu/Co trilayers when current flows perpendicular to the plane. We show that resistance of the Co/Cu interface depends on the proximity of another interface, which makes the parameters characterizing the spin-dependent interface resistance ARF/N* and γF/N, to be dependent on the layer thickness separating the two interfaces. This leads to a decrease in the measurable quantity SR=√(ARAP)(ARAP-ARP) with the Cu layer thickness and, therefore, to the departure from the series-resistor model. Here ARAP is the specific resistance (area A times resistance R) of the trilayer when magnetizations of the two Co layers are aligned antiparallel (AP) to each other, and ARP is the specific resistance when the layer magnetizations are aligned parallel (P). We demonstrate that recent experimental data on currentperpendicular-to-plane transport in Co/Cu/Co spin valves can be explained by the interface proximity effects without introducing a finite spin-diffusion length.