Strong form of Hensel lemma

Let $K$ be complete for a discrete valuation $v$. If $f\in {\mathcal{O}}_K[X]$ andthen Newton iteration over a valued field converges to a root $\alpha \in {\mathcal{O}}_K$ with $v(\alpha -a)>v(f^{\prime }(a))$.

Starting from $a_0$, Newton iteration sets $a_{n+1}=a_n-f(a_n)/f^{\prime }(a_n)$. Under the strong form of Hensel lemma inequality, the correction valuations grow at least geometrically, so $(a_n)$ is a Cauchy sequence converging to a root.