Active Phase for Activated Random Walk on Z

We consider the Activated Random Walk model on $mathbb{Z}$. In this model, each particle performs a continuous-time simple symmetric random walk, and falls asleep at rate $lambda$. A sleeping particle does not move but it is reactivated in the presence of another particle. We show that for any sleep rate $lambda < infty$ if the density $ zeta $ is close enough to $1$ then the system stays active.