Lemma 15.11.4. Let $(A, I)$ be a pair. If $A$ is $I$-adically complete, then the pair is henselian.

**Proof.**
By Algebra, Lemma 10.96.6 the ideal $I$ is contained in the Jacobson radical of $A$. Let $f \in A[T]$ be a monic polynomial and let $\overline{f} = g_0h_0$ be a factorization of $\overline{f} = f \bmod I$ with $g_0, h_0 \in A/I[T]$ monic generating the unit ideal in $A/I[T]$. By Lemma 15.11.2 we can successively lift this factorization to $f \bmod I^ n = g_ n h_ n$ with $g_ n, h_ n$ monic in $A/I^ n[T]$ for all $n \geq 1$. As $A = \mathop{\mathrm{lim}}\nolimits A/I^ n$ this finishes the proof.
$\square$

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