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

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## 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.): $\S\text I.3$