I have a question about Faltings' paper "Crystalline cohomology and p-adic Galois representations". Suppose $R$ is a ring such that there is an etale map $\mathbb{Z}_{p}[T,T^{-1}]\to R$.

By $\bar{R}$ we denote the maximal extension of $R$ which is etale in characteristic zero.

The paper states that the Frobenius map on $\bar{R}/p\bar{R}$ is surjective. I wonder why. Is there any reference for the proof? Thank you!