Hausdorff's Maximal Principle implies Zorn's Lemma

Theorem
The Hausdorff Maximal Principle implies Zorn's Lemma.

Proof
Let $\struct {\PP, \preceq}$ be a partially ordered set.

The Hausdorff Maximal Principle states that there is a maximal chain in $\struct {\PP, \preceq}$.

If this maximal chain has an upper bound in $\PP$, then such an upper bound is a maximal element of $\PP$ itself, by the maximality of the chain.