A note on growth of hyperbolic groups

The following short note provides an alternative proof of a result of Coornaert: namely, that given a non-elementary word-hyperbolic group $G$ with a finite generating set $X$, there exist constants $\lambda ,D>1$ such that $$D^{-1}{\lambda }^n\le \mathrm{\mid }{B}_{G,X}(n)\mathrm{\mid }\le D{\lambda }^n$$ for all $n\ge 0$, where $B_{G,X}(n)$ is the ball of radius $n$ in the Cayley graph $\mathrm{\Gamma }(G,X)$.