Soon after the detection by the Compton Gamma Ray Observatory of gamma-rays above 100 MeV from the optically violent variable 3C 279 at a redshift of z = 0.54, Stecker et al. pointed out that TeV gamma-rays from large-redshift sources would be attenuated by photon-photon interactions in the intergalactic infrared (IR) background radiation. In this paper, we consider the case in which the produced pairs inverse-Compton-scatter low-energy photons up to gamma-ray energies and start a pair-Compton cascade in the IR/microwave background. The simulation of such cascades over the enormous distances from high-redshift sources required us to develop a new matrix method to carry out the propagation. We find that the TeV gamma-ray fluxes are only marginally higher for the case with cascading than they are for the case of absorption only. However, the greatest difference occurs at lower energies, around 1 GeV, where the pile-up of photons below the absorption feature results in a significant flattening in the observed spectrum relative to the emission spectrum.