Every finite set of integers is an asymptotic approximate group

A set $A$ is an $(r,\mathrm{\ell })$-approximate group in the additive abelian group $G$ if $A$ is a nonempty subset of $G$ and there exists a subset $X$ of $G$ such that $\mathrm{\mid }X\mathrm{\mid }\le \mathrm{\ell }$ and $rA\subseteq X+A$. The set $A$ is an asymptotic $(r,\mathrm{\ell })$-approximate group if the sumset $hA$ is an $(r,\mathrm{\ell })$-approximate group for all sufficiently large integers $h$. It is proved that every finite set of integers is an asymptotic $(r,r+1)$-approximate group for every integer $r\ge 2$.