Greatest Element is Maximal

Theorem
Let $\struct {S, \preceq}$ be an ordered set which has a greatest element.

Let $M$ be the greatest element of $\struct {S, \preceq}$.

Then $M$ is a maximal element.

Proof
By definition of greatest element:
 * $\forall y \in S: y \preceq M$

Suppose $M \preceq y$.

As $\preceq$ is an ordering, $\preceq$ is by definition antisymmetric.

Thus it follows by definition of antisymmetry that $M = y$.

That is:
 * $M \preceq y \implies M = y$

which is precisely the definition of a maximal element.

Also see

 * Smallest Element is Minimal