Hulanicki-Reiter condition states:
A finetely generated group G is amenable if and only if for every $\epsilon>0$ and $R>0$ there exists a function $f\in l_1(G)_{1,+}$ such that:
a) $||f-\gamma f||_1<\epsilon$ for all $\gamma\in G$ when $|\gamma|\leq R$
b) $\text{supp}f$ is finite
where $l_1(G)_{1,+}=\{f\in l_1(G):||f||_1=1\text{ and }f\geq 0$
Using this condition show that free group $\mathbb{F}_2$ is not amenable.
I only know proof of nonamenability of free group by use of invariant mean, but I heard that there is somewhere proof of this fact which only uses Hulanicki-Reiter condition, but I neither can find it nor prove it by myself.