= Prime-power contraction in a formal group
{title2=$[p^n](x)\longrightarrow0\quad(x\in\mathfrak m)$}
Let $R$ be the <valuation ring> of a complete <non-Archimedean valued field> whose residue field has characteristic $p$, and let $\mathfrak m=\{x:|x|<1\}$. For a one-dimensional <formal group law> over $R$, repeated addition gives $[p](X)=pX+\sum_{j\ge2}c_jX^j$ with $c_j\in R$. Hence $|[p](x)|\le\max(|p||x|,|x|^2)$. The series converges for $x\in\mathfrak m$ because its <coefficients>[coefficients] are bounded and $|x|<1$.
For $x_0=x$ and $x_{n+1}=[p](x_n)$, choose $\rho=\max(|p|,|x|)<1$. The values remain in the ball $|x_n|\le|x|$, so $|x_n|\le\rho^n|x|\to0$. Associativity identifies $x_n=[p^n](x)$. The argument also works in equal characteristic $p$, where the linear <coefficient>[coefficient] vanishes, and does not require a <discrete valuation>. It proves contraction of <prime number>[prime]-power multiplication near the formal identity without assuming a characteristic-zero formal logarithm.
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