  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.
