Every finite subset of an abelian group is an asymptotic approximate group

If A is a nonempty subset of an additive abelian group G, then the h-fold sumset ishA={x1+⋯+xh:xi∈Ai for i=1,2,…,h}. We do not assume that A contains the identity, nor that A is symmetric, nor that A is finite. The set A is an (r,ℓ)-approximate group in G if there exists a subset X of G such that |X|≤ℓ and rA⊆XA. The set A is an asymptotic(r,ℓ)-approximate group if the sumset hA is an (r,ℓ)-approximate group for all sufficiently large h. It is proved that every polytope in a real vector space is an asymptotic (r,ℓ)-approximate group, that every finite set of lattice points is an asymptotic (r,ℓ)-approximate group, and that every finite subset of every abelian group is an asymptotic (r,ℓ)-approximate group.