Why is the Frobenius morphism for Witt rings epimorphism?

Question

In Demazure's book ''lectures on $p$-divisible groups'' page 57, he said the Frobenius morphism $F:W_k\to W_k$ is given by $$F(a_0,\cdots, a_n,\cdots)=(a_0^p,\cdots,a_n^p,\cdots)$$ and in the proof of the proposition he said $F$ is an epimorphism. But it seems that we need $k$ to be perfect here, right?