 Simple-root Hensel lifting by successive residues

For a complete discrete valuation ring $R$, let $f(a_1)\in (\pi )$ and let $f^{\prime }(a_1)$ be a unit. If $f(a_n)\in ({\pi }^n)$, choose $c$ modulo $\pi$ solving $f(a_n)/{\pi }^n+c{f}^{\prime }(a_n)\equiv 0$ and put $a_{n+1}=a_n+c{\pi }^n$. Taylor expansion gives $f(a_{n+1})\in ({\pi }^{n+1})$. The sequence is Cauchy and its limit is a root. If two roots lift the same residue, factoring $f(x)-f(y)$ leaves $x-y$ times a unit, proving uniqueness.