Zorn's Lemma

Theorem
Let $X$ be a non-empty poset such that every non-empty chain in $X$ has an upper bound in $X$.

Then $X$ has at least one maximal element.

Proof
By the Hausdorff Maximal Principle there is a maximal chain $C \subseteq X$. Let $c$ be an upper bound for this chain (which must exist by hypothesis of the lemma). $c$ must be maxim

al, for if $d > c$ then $C \cup \{d\}$ would be a chain strictly including $C$. QED.