Exercise 4, § 1

Let $A$ be a ring, $a = (a_n)_{n \in \mathbf{N}}$ an element of $W(A)$, $a^{(p)}$ the element $(a_n^p)_{n \in \mathbf{N}}$ of $W(A)$. Put

Prove that one has $b_n - pa_n \in p^{p-1}A$ for every $n \in \mathbf{N}$. (Reduce to the case where $A = \mathbf{Z}[(X_n)_{n \in \mathbf{N}}]$ and $a = (X_n)_{n \in \mathbf{N}}$ and use Prop. 1 of § 1, No. 1.)