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

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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.


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