  A packing by a body $K$ is collection of congruent copies of $K$ (in either
Euclidean or hyperbolic space) so that no two copies intersect nontrivially in
their interiors. A covering by $K$ is a collection of congruent copies of $K$
such that for every point $p$ in the space there is copy in the collection
containing $p$. A completely saturated packing is one in which it is not
possible to replace a finite number of bodies of the packing with a larger
number and still remain a packing. A completely reduced covering is one in
which it is not possible to replace a finite number of bodies of the covering
with a smaller number and still remain a covering. It was conjectured by G.
Fejes Toth, G. Kuperberg, and W. Kuperberg that completely saturated packings
and commpletely reduced coverings exist for every body $K$ in either
$n$-dimensional Euclidean or $n$-dimensional hyperbolic space. We prove this
conjecture.
