Hausdorff's Maximal Principle implies Zorn's Lemma

Theorem
The Hausdorff Maximal Principle implies Zorn's Lemma.

Proof
Let $\left({\mathcal P, \preceq}\right)$ be a partially ordered set.

The Hausdorff Maximal Principle states that there is a maximal chain in $\left({\mathcal P, \preceq}\right)$.

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