Supremum of Absolute Value of Difference equals Difference between Supremum and Infimum

Theorem
Let $f$ be a real function.

Let $S$ be a subset of the domain of $f$.

Let $\displaystyle \sup_{x \mathop \in S} \set {\map f x}$ and $\displaystyle \inf_{x \mathop \in S} \set {\map f x}$ exist.

Then $\displaystyle \sup_{x, y \mathop \in S} \set {\size {\map f x - \map f y} }$ exists and:


 * $\displaystyle \sup_{x, y \mathop \in S} \set {\size {\map f x - \map f y} } = \sup_{x \mathop \in S} \set {\map f x} - \inf_{x \mathop \in S} \set {\map f x}$