Interval Defined by Absolute Value

From ProofWiki

Jump to: navigation, search

Theorem

Let $\xi, \delta \in \R$ be real numbers.

Let $\delta > 0$.

Then:

$\left\{{x \in \R: \left\vert{\xi - x}\right\vert < \delta}\right\} = \left({\xi - \delta .. \xi + \delta}\right)$

where $\left({\xi - \delta .. \xi + \delta}\right)$ is the open real interval between $\xi - \delta$ and $\xi + \delta$.

Similarly:

$\left\{{x \in \R: \left\vert{\xi - x}\right\vert \le \delta}\right\} = \left[{\xi - \delta .. \xi + \delta}\right]$

where $\left[{\xi - \delta .. \xi + \delta}\right]$ is the closed real interval between $\xi - \delta$ and $\xi + \delta$.

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \left\vert{\xi - x}\right\vert$	$<$	$\displaystyle \delta$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle -\delta$	$<$	$\displaystyle \xi - x < \delta$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Corollary to negative of absolute value
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle \delta$	$>$	$\displaystyle x - \xi - x > -\delta$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle \xi + \delta$	$>$	$\displaystyle x > \xi - \delta$	$\displaystyle $	$\displaystyle $	$\displaystyle $

But:

$\left({\xi - \delta .. \xi + \delta}\right) = \left\{{x \in \R: \xi - \delta < x < \xi + \delta}\right\}$

The other result follows similarly.

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left\vert{\xi - x}\right\vert\)	\(<\)	\(\displaystyle \delta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle -\delta\)	\(<\)	\(\displaystyle \xi - x < \delta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Corollary to negative of absolute value
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle \delta\)	\(>\)	\(\displaystyle x - \xi - x > -\delta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle \xi + \delta\)	\(>\)	\(\displaystyle x > \xi - \delta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)