Finite Non-Empty Subset of Totally Ordered Set has Smallest and Greatest Elements/Proof 2

Theorem

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

Then every finite $T$ such that $\O \subset T \subseteq S$ has both a smallest and a greatest element.

Proof

The result follows from:

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

Minimal Element of Chain is Smallest Element

Maximal Element of Chain is Greatest Element

$\blacksquare$

Sources

• 1967: Garrett Birkhoff: Lattice Theory (3rd ed.): $\S\text I.3$