On the existence of completely saturated packings and completely reduced coverings

We prove the following conjecture of G. Fejes Toth, G. Kuperberg, and W. Kuperberg: every body Kin either n-dimensional Euclidean or n-dimensional hyperbolic space admits a completely saturated packing and a completely reduced covering. Also we prove the following counterintuitive result: for every epsilon > 0, there is a body K in hyperbolic n-space which admits a completely saturated packing with density less than epsilon but which also admits a tiling.