Hausdorff's Maximal Principle implies Axiom of Choice/Proof

Proof
Let Hausdorff's Maximal Principle be accepted.

Let $S$ be a non-empty set.

Let $\Sigma$ be the set of all choice functions of all non-empty subsets of $S$.

From Countable Set has Choice Function every finite subset of $S$ has a choice function.

Hence $\Sigma$ is a non-empty set.

Lemma
By Hausdorff's Maximal Principle it follows that $\Sigma$ has a maximal element $f$.

Hence $f$ is seen to be a choice function for $S$.