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 $\displaystyle \sup_{x \mathop \in S} \left\{{f \left({x}\right)}\right\}$ and $\displaystyle \inf_{x \mathop \in S} \left\{{f \left({x}\right)}\right\}$ exist.


Then $\displaystyle \sup_{x, y \mathop \in S} \left\{{\left\vert{f \left({x}\right) - f \left({y}\right)}\right\vert}\right\}$ exists and:

$\displaystyle \sup_{x, y \mathop \in S} \left\{{\left\vert{f \left({x}\right) - f \left({y}\right)}\right\vert}\right\} = \sup_{x \mathop \in S} \left\{{f \left({x}\right)}\right\} - \inf_{x \mathop \in S} \left\{{f \left({x}\right)}\right\}$


Proof

\(\displaystyle \sup_{x \mathop \in S} \left\{ {f \left({x}\right)}\right\} - \inf_{x \mathop \in S} \left\{ {f \left({x}\right)}\right\}\) \(=\) \(\displaystyle \sup_{x \mathop \in S} \left\{ {f \left({x}\right)}\right\} + \sup_{x \mathop \in S} \left\{ {- f \left({x}\right)}\right\}\) Negative of Infimum is Supremum of Negatives
\(\displaystyle \) \(=\) \(\displaystyle \sup_{x, y \mathop \in S} \left\{ {f \left({x}\right) + \left({- f \left({y}\right)}\right)}\right\}\) Supremum of Sum equals Sum of Suprema
\(\displaystyle \) \(=\) \(\displaystyle \sup_{x, y \mathop \in S} \left\{ {f \left({x}\right) - f \left({y}\right)}\right\}\)
\(\displaystyle \) \(=\) \(\displaystyle \sup_{x, y \mathop \in S} \left\{ {\left\vert{f \left({x}\right) - f \left({y}\right)}\right\vert}\right\}\) Supremum of Absolute Value of Difference equals Supremum of Difference

$\blacksquare$