About the definition of amenable group. An amenable group is at most contable group $G$ for which exist a sequence $\{F_n\}_{n\in \mathbb{N}}$ of finite sets such that
$$\displaystyle\lim_{n\rightarrow \infty}\frac{|(g\cdot F_n)\triangle F_n|}{|F_n|}=0 $$
for every $g\in G$.
There is something that I am not understanding about this definition. The "almost trvial" example is $\mathbb{Z}$. Consider the sequence of sets of the form $[-N,N]=\{-N,-N+1,...,N-1, N\}$.
I tried to prove that indeed this sequence shows that $\mathbb{Z}$ is amenable. 
But if we do the calculation $[-N,N]\triangle [-N+a,N+a]|=|[-N,N]\cup [-N+a,N+a]|=4N$ for every $a$ big enough and $|[-N,N]|=2N$ but this implies that
$$\displaystyle\lim_{n\rightarrow \infty}\frac{|[-N,N]\cup [-N+a,N+a]|}{|[-N,N]|}=2 $$
What am I doing wrong?
Thanks!

If you set $F_N := [-N, N]$, then for big enough $N$ you will have that $a F_N \Delta F_N$ has $2 |a|$ elements ($a$ is fixed!!), while $F_N$ has always $2N + 1$ elements (don't forget to count $0$). So for $N \to \infty$ this goes to $0$ as desired.
