Real-Valued Function on Finite Set is Bounded

Theorem
Let $S$ be a finite set.

Let $f: S \to \R$ be a real-valued function on $S$.

Then $f$ is bounded.

Proof
Let $K$ be defined as:
 * $K = \ds \max_{x \mathop \in S} \size {\map f x}$

where $\size {\map f x}$ denotes the absolute value of $\map f x$.

Then trivially:
 * $\exists K \in \R_{\ge 0}: \forall x \in S: \size {\map f x} \le K$

This is the definition of a bounded real-valued function.