Maximal Element of Chain is Greatest Element

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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 $C$.

$\blacksquare$


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