# 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:

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

## Proof

 $\displaystyle \sup_{x \mathop \in S} \set {\map f x} - \inf_{x \mathop \in S} \set {\map f x}$ $=$ $\displaystyle \sup_{x \mathop \in S} \set {\map f x} + \sup_{x \mathop \in S} \set {-\map f x}$ Negative of Infimum is Supremum of Negatives $\displaystyle$ $=$ $\displaystyle \sup_{x, y \mathop \in S} \set {\map f x + \paren {-\map f y} }$ Supremum of Sum equals Sum of Suprema $\displaystyle$ $=$ $\displaystyle \sup_{x, y \mathop \in S} \set {\map f x - \map f y}$ $\displaystyle$ $=$ $\displaystyle \sup_{x, y \mathop \in S} \set {\size {\map f x - \map f y} }$ Supremum of Absolute Value of Difference equals Supremum of Difference

$\blacksquare$