Open Interval Defined by Absolute Value

Theorem

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

Let $\delta > 0$.

Then:

$\set {x \in \R: \size {\xi - x} < \delta} = \openint {\xi - \delta} {\xi + \delta}$

where $\openint {\xi - \delta} {\xi + \delta}$ is the open real interval between $\xi - \delta$ and $\xi + \delta$.

Proof

\(\ds \size {\xi - x}\)

\(<\)

\(\ds \delta\)

\(\ds \leadstoandfrom \ \ \)

\(\ds -\delta\)

\(<\)

\(\ds \xi - x < \delta\)

Negative of Absolute Value: Corollary 1

\(\ds \leadstoandfrom \ \ \)

\(\ds \delta\)

\(>\)

\(\ds x - \xi > -\delta\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \xi + \delta\)

\(>\)

\(\ds x > \xi - \delta\)

But:

$\openint {\xi - \delta} {\xi + \delta} = \set {x \in \R: \xi - \delta < x < \xi + \delta}$

$\blacksquare$

Also presented as

$\set {x \in \R: \size {x - \xi} < \delta} = \openint {\xi - \delta} {\xi + \delta}$

which is immediate from:

$\size {x - \xi} = \size {\xi - x}$

Also see

• Closed Interval Defined by Absolute Value

• Complement of Open Interval Defined by Absolute Value
• Complement of Closed Interval Defined by Absolute Value

Sources

• 1973: G. Stephenson: Mathematical Methods for Science Students (2nd ed.) ... (previous) ... (next): Chapter $1$: Real Numbers and Functions of a Real Variable: $1.2$ Operations with Real Numbers
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 2$: Continuum Property: Exercise $\S 2.10 \ (2)$