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

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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

$\blacksquare$