Finite Non-Empty Subset of Ordered Set has Maximal and Minimal Elements/Corollary

Theorem
Let $\struct{S, \preceq}$ be a finite ordered set.

Let $x \in S$.

Then there exists a maximal element $M \in S$ and a minimal element $m \in S$ such that:
 * $m \preceq x \preceq M$

Proof
Let $T = \set{y : x \preceq y}$.

By the reflexivity of the ordering $\preceq$:
 * $x \preceq x$

So $x \in T$ and $T$ is non-empty.

From Finite Non-Empty Subset of Ordered Set has Maximal and Minimal Elements:
 * $\struct{T, \preceq}$ has a maximal element $M \in T$

We now show that $M$ is a maximal element in $\struct{S, \preceq}$.

Let $y \in S$ such that:
 * $M \preceq y$

By the transitiviy of the ordering $\preceq$:
 * $x \prec y$

So $y \in T$.

By the definition of a maximal element:
 * $y = M$

Similarly for $T' = \set{y : y \preceq x}$:
 * $\struct{T', \preceq}$ has a minimal element $m \in T'$

and $m$ is a minimal element in $\struct{S, \preceq}$