Hausdorff's Maximal Principle implies Zorn's Lemma

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Theorem

Hausdorff's Maximal Principle implies Zorn's Lemma.

Proof

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

Hausdorff's 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.

$\blacksquare$

Sources

• 1984: Gerald B. Folland: Real Analysis: Modern Techniques and their Applications : (P.2)