Maximal Element of Chain is Greatest Element

Theorem
Let $\left({S, \preceq}\right)$ be an ordered set.

Let $C$ be a chain in $S$.

Let $m$ be a maximal element of $C$.

Then $m$ is the greatest element of $C$.

Proof
Let $x \in C$.

Since $m$ is maximal in $C$, $m \not\prec x$.

Since $C$ is a chain, $x = m$ or $x \prec m$.

Thus for each $x \in C$, $x \preceq m$.

Therefore $m$ is the greatest element of $C$.

Also see

 * Minimal Element of Chain is Smallest Element