Distance from Subset of Real Numbers

Theorem

Let $S$ be a subset of the set of real numbers $\R$.

Let $x \in \R$ be a real number.

Let $\map d {x, S}$ be the distance between $x$ and $S$.

Then:

Distance from Subset of Real Numbers to Element

$x \in S \implies \map d {x, S} = 0$

Distance from Subset of Real Numbers to Supremum

Let $S$ be bounded above such that $\xi = \sup S$.

Then:

$\map d {\xi, S} = 0$

Distance from Subset of Real Numbers to Infimum

Let $S$ be bounded below such that $\xi = \inf S$.

Then:

$\map d {\xi, S} = 0$

Real Number at Distance Zero from Closed Real Interval is In Interval

Let $I \subseteq \R$ be a closed real interval.

Then:

$\map d {x, I} = 0 \implies x \in I$

Existence of Real Number at Distance Zero from Open Real Interval not in Interval

Let $I \subseteq \R$ be an open real interval such that $I \ne \O$ and $I \ne \R$.

Then:

$\exists x \notin I: \map d {x, I} = 0$

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 2$: Continuum Property: Exercise $\S 2.13 \ (5)$