Supremum of Absolute Value of Difference equals Supremum of Difference
Theorem
Let $S$ be a non-empty real set.
Let $\ds \sup_{x, y \mathop \in S} \paren {x - y}$ exist.
Then $\ds \sup_{x, y \mathop \in S} \size {x - y}$ exists and:
- $\ds \sup_{x, y \mathop \in S} \size {x - y} = \sup_{x, y \mathop \in S} \paren {x - y}$
Proof
Consider the set $\set {x - y: x, y \in S, x - y \le 0}$.
There is a number $x'$ in $S$ as $S$ is non-empty.
Therefore, $0 \in \set {x - y: x, y \in S, x - y \le 0}$ as $x = y = x'$ implies that $x - y = 0$, $x, y \in S$, and $x - y \le 0$.
Also, $0$ is an upper bound for $\set {x - y: x, y \in S, x - y \le 0}$ by definition.
Accordingly:
- $\ds \sup_{x, y \mathop \in S, x − y \mathop \le 0} \paren {x - y} = 0$
Consider the set $\left\{{x - y: x, y \in S, x - y \ge 0}\right\}$.
There is a number $x'$ in $S$ as $S$ is non-empty.
Therefore, $0 \in \left\{{x - y: x, y \in S, x - y \ge 0}\right\}$ as $x = y = x'$ implies that $x - y = 0$, $x, y \in S$, $x - y \ge 0$.
Accordingly:
- $\ds \sup_{x, y \mathop \in S, x − y \mathop \ge 0} \paren {x - y} \ge 0$
![]() | Although this article appears correct, it's inelegant. There has to be a better way of doing it. In particular: I can't immediately think of how it would be done, but it would be good if we could devise a neater and more compact notation that what is used here. All the complicated mathematics is being done in the underscript, which makes it not easy to follow. (Improved Dec. 2016.) You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Improve}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
\(\ds \sup_{x, y \mathop \in S} \paren {x - y}\) | \(=\) | \(\ds \sup_{x, y \mathop \in S, x − y \mathop \ge 0 \text { or } x − y \mathop \le 0} \paren {x - y}\) | as ($x - y \ge 0$ or $x - y \le 0$) is true | |||||||||||
\(\ds \) | \(=\) | \(\ds \max \set {\sup_{x, y \mathop \in S, x − y \mathop \ge 0} \paren {x - y}, \sup_{x, y \mathop \in S, x − y \mathop \le 0} \paren {x - y} }\) | by Supremum of Set Equals Maximum of Suprema of Subsets | |||||||||||
\(\ds \) | \(=\) | \(\ds \max \set {\sup_{x, y \mathop \in S, x − y \mathop \ge 0} \paren {x - y}, 0}\) | as $\ds \sup_{x, y \mathop \in S, x − y \mathop \le 0} \paren {x - y} = 0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sup_{x, y \mathop \in S, x − y \mathop \ge 0} \paren {x - y}\) | as $\ds \sup_{x, y \mathop \in S, x − y \mathop \ge 0} \paren {x - y} \ge 0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sup_{x, y \mathop \in S, x − y \mathop \ge 0} \size {x - y}\) | as $\size {x − y} = x − y$ since $x − y \ge 0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \max \set {\sup_{x, y \mathop \in S, x − y \mathop \ge 0} \size {x - y}, \sup_{x, y \mathop \in S, x − y \mathop \ge 0} \size {x - y} }\) | as the two arguments of max are equal | |||||||||||
\(\ds \) | \(=\) | \(\ds \max \set {\sup_{x, y \mathop \in S, x − y \mathop \ge 0} \size {x - y}, \sup_{y, x \mathop \in S, y − x \mathop \ge 0} \size {y - x} }\) | by renaming variables $x \leftrightarrow y$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \max \set {\sup_{x, y \mathop \in S, x − y \mathop \ge 0} \size {x - y}, \sup_{x, y \mathop \in S, x − y \mathop \le 0} \size {x - y} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sup_{x, y \mathop \in S, x − y \mathop \ge 0 \text { or } x − y \mathop \le 0} \size {x - y}\) | by Supremum of Set Equals Maximum of Suprema of Subsets | |||||||||||
\(\ds \) | \(=\) | \(\ds \sup_{x, y \mathop \in S} \size {x - y}\) | as ($x - y \ge 0$ or $x - y \le 0$) is true |
$\blacksquare$