FALSE: every Dedekind cut has a greatest element

(C1, C2) $0^\ast = \{ q \in \mathbb{Q} : q < 0 \}$ is nonempty ($-1 \in 0^\ast$) and proper ($0 \notin 0^\ast$), and it is downward closed: $q < p < 0 \Rightarrow q < 0$, so $q \in 0^\ast$.

(C3) For any $q \in 0^\ast$, i.e. $q < 0$, the rational $q/2$ satisfies $q < q/2 < 0$, so $q/2 \in 0^\ast$ and $q/2 > q$; hence no element of $0^\ast$ is greatest.

By steps 1.1 and 1.2, $0^\ast$ satisfies (C1)–(C3): it is a Dedekind cut (Dedekind cut), i.e. a real number (The real numbers $\mathbb{R}$ as Dedekind cuts), with no greatest element.

Thus $0^\ast$ is a Dedekind cut having no greatest element, directly contradicting the claim; more strongly, "no greatest element" is precisely axiom (C3), which every cut must satisfy, so the claim fails for all cuts and is false.