A topological group $G$ is said to satisfy the Leptin condition if for every compact subset $K\subseteq G$ and for every $\epsilon>0$ there exists a compact subset $L$ such that

$\mu(LK)$ < $ (1+\epsilon)\mu(L)$

where $\mu$ is Haar measure (I'm assuming $G$ locally compact hausdorff). It is known that the Leptin condition is equivalent to amenability, a very different looking property concerning the existence of an invariant mean on the group. How can one prove that nilpotent lie groups satisfy the Leptin condition without passing through amenability? For example $\mathbb{R}^n$ almost trivially satisfies such condition, even if proving amenability is harder. Thanks in advance!