Proposition 4

Let $a$ and $b$ be in $A$ and let $x = (x_n)_{n \in \mathbf{N}}$ be an element of $W(A)$.

a) We have the formulas

b) The series with general term $V^n(\tau(x_n))$ is convergent in $W(A)$, with sum $x$. Let $n$ be a positive integer. The polynomial $P_n(X_0, ..., X_n; Y_0, ..., Y_n)$ introduced in No. 3 is isobaric of weight $p^n$ in the family $(X_0, ..., X_n)$ when $X_i$ is assigned the weight $p^i$. Hence

Since $1 = (1, 0, 0, ...)$ is the unit element of the ring of Witt vectors with coefficients in $\mathbf{Z}[(X_n)_{n \in \mathbf{N}}, (Y_n)_{n \in \mathbf{N}}]$, we have By substituting $a$ for $X_0$ and $x_i$ for $Y_i$, we deduce from (40) and (41) the relation: According to the definition of the multiplication in $W(A)$, we have proved (39); formula (38) is a particular case of (39).

Let us prove b). By definition, $V^n(\tau(x_n))$ is the sequence all of whose components are zero, except the one with index $n$, which is equal to $x_n$. It follows from Lemma 4, by induction on $m$, that we have

for every integer $m \geq 0$; we deduce b) by passing to the limit since the topology $\mathscr{T}$ on $W(A)$ is the product of the discrete topologies of the factors $A$.