For a $p$-adic field $K$ with perfect residue field $k$, we know the standard construction of the ring $R$. I will recall it briefly. It is $\varprojlim_{x \rightsquigarrow x^p} O_{C_K}/pO_{C_K}$, with component-wise addition. It turns out that $R$ is a perfect ring of characteristic $p$ that is also a complete (non-discrete) valuation ring, and let us denote the valuation map by $v_R$. The residue field of $R$ is the algebraic closure $\overline{k}$ of the residue field $k$ of $K$. We can form the ring $W(R)$ of Witt vectors of $R$, and we can form the ring $W(\textrm{fof}(R))$ of Witt vectors of the field of fractions $\textrm{fof}(R)$ of $R$. It is a known fact that $\textrm{fof}(R)$ is algebraically closed. As $\textrm{fof}(R)$ is a perfect field of positive characteristic, $W(\textrm{fof}(R))$ is a d.v.r. with $p$ as a generator for the maximal ideal. But $R$ is not a field, so $W(R)$ is not a d.v.r., but is complete with respect to the topology generated by powers of $p$. But as the construction of $W$ is functorial, there is an inclusion map $W(R) \hookrightarrow W(\textrm{fof}(R))$.
1) What is the topology on $W(R)$? Is it the subspace topology from $W(\textrm{fof}(R))$? I guess not, as the $p$-adic valuation on $W(\textrm{fof}(R))$ "forgets" $v_R$, that is to say every nonzero element in $\textrm{fof}(R)$ and therefore every nonzero element of $R$ becomes a unit in $W(\textrm{fof}(R))$, but not necessarily in $W(R)$.
2)Let $\varepsilon$ be the element of $R$ given by the sequence $(\zeta_{p^n}-1 \mod p)$. Consider the Teichmuller lift $[\varepsilon] \in W(R)$ and set $\pi_\varepsilon := [\varepsilon]-1$. Now the claim is that power series in $\pi_\varepsilon$ with, say, integer coefficients converge in $W(R)$. (This is an unchecked claim in Fontaine's book, on page 79). Since $W(R)$ is not a valuation ring, I thought of showing that $\pi_{\varepsilon}$, or at least a power of it is in $pW(R)$. We observe that $\pi_{\varepsilon}=(\varepsilon-1, \ldots)$ as an element in $W(R)$, but an element of $pW(R)$ has the first coordinate $0$, which in this case, $\varepsilon-1$ is not. What am I missing? Actually I can show that $\pi_{\varepsilon} \in pW(R)$ by making use of the map $\theta: W(R)\to O_{C_K}$, but since this is all so confusing, I would like to know where I went wrong above.