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

Theorem
Let $\left({S, \preceq}\right)$ be a totally ordered set.

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

Proof
The result follows from:


 * Finite Nonempty Subset of Ordered Set has Maximal and Minimal Elements
 * Minimal Element of Chain is Smallest Element
 * Maximal Element of Chain is Greatest Element