Distance from Subset of Real Numbers

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Theorem

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

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

Let $d \left({x, S}\right)$ be the distance between $x$ and $S$.


Then:

$(1): \quad x \in S \implies d \left({x, S}\right) = 0$
$(2): \quad$ If $S$ is bounded above and $x = \sup S$, then $d \left({x, S}\right) = 0$
$(3): \quad$ Similarly, if $S$ is bounded below and $x = \inf S$, then $d \left({x, S}\right) = 0$
$(4): \quad$ If $I$ is a closed real interval, then $d \left({x, I}\right) = 0 \implies x \in I$
$(5): \quad$ If $I$ is an open real interval apart from $\varnothing$ or $\R$, then $\exists x \notin I: d \left({x, I}\right) = 0$.


Proof

From the definition of distance:

$\forall x, y \in \R: d \left({x, y}\right) = \left|{x - y}\right|$

Thus:

$\displaystyle d \left({x, S}\right) = \inf_{y \mathop \in S} \left({\left|{x - y}\right|}\right)$


$x \in S \implies d \left({x, S}\right) = 0$:

Consider the set $T = \left\{{\left|{x - y}\right|: y \in S}\right\}$.

This has $0$ as a lower bound as Absolute Value is Bounded Below by Zero.

So:

$\displaystyle d \left({x, S}\right) = \inf_{y \mathop \in S} \left({\left|{x - y}\right|}\right) \ge 0$

If $x \in S$ then:

$\left|{x - x}\right| = 0 \in T$

and so:

$\displaystyle 0 \le \inf_{y \mathop \in S} \left({d \left({x, y}\right)}\right)$

Thus:

$\displaystyle d \left({x, S}\right) = \inf_{y \mathop \in S} \left({d \left({x, y}\right)}\right) = 0$

$\blacksquare$


If $S$ is bounded above and $x = \sup S$, then $d \left({x, S}\right) = 0$:

Let $x = \sup S$.

Then:

$\forall y \in S: \left|{x - y}\right| = x - y$

So we need to show that no $h > 0$ can be a lower bound for $T = \left\{{\left|{x - y}\right|: y \in S}\right\}$.

Suppose this is false, and $\exists h > 0: \forall y \in S: x - y \ge h$.

But then:

$\forall y \in S: y \le x - h$

and hence $x - h$ is an upper bound for $T$ smaller than $x = \sup S$.

But this is supposed to be the supremum, that is the smallest upper bound.

So there is no such $h > 0$ and so $d \left({x, S}\right) = 0$.

$\blacksquare$


If $S$ is bounded below and $x = \inf S$, then $d \left({x, S}\right) = 0$:

Consider $d \left({-x, S'}\right)$ where $S' = \left\{{-x: x \in S}\right\}$.

By Negative of Infimum is Supremum of Negatives, $x = \inf S \implies-x = \sup S'$.

Thus from the above, $d \left({-x, S'}\right) = 0$ and hence the result.

$\blacksquare$


If $I$ is a closed real interval, then $d \left({x, I}\right) = 0 \implies x \in I$:

Since $I$ is an interval, if $x \notin I$ then $x$ is either an upper bound or a lower bound for $I$.

Suppose $x$ is an upper bound for $I$.

Let $B$ be the supremum of $I$.

Then because $I$ is closed, $B \in I$.


So:

\(\displaystyle \forall y \in I:\) \(\displaystyle \left\vert{x - y}\right\vert\) \(=\) \(\displaystyle x - y\)                    
\(\displaystyle \) \(=\) \(\displaystyle x - B + B - y\)                    
\(\displaystyle \) \(=\) \(\displaystyle x - B + \left\vert{B - y}\right\vert\)                    

Now from Infimum Plus Constant:

$\inf_{y \in S} \left|{x - y}\right| = x - B + \inf_{y \in S} \left|{B - y}\right|$

But we also have:

$x - B \ge 0$
$d \left({B, S}\right) \ge 0$
$d \left({x, S}\right) = 0$

So it follows that $x = B$ and so $x \in I$.


A similar argument applies if $x$ is a lower bound for $I$.

$\blacksquare$


If $I$ is an open real interval apart from $\varnothing$ or $\R$, then $\exists x \notin I: d \left({x, I}\right) = 0$:

As $I \ne \varnothing$ and $I \ne \R$ it follows that one of the following applies:

$\exists a, b \in \R: I = \left({a \,.\,.\, b}\right)$
$\exists a \in \R: I = \left({a \,.\,.\, \infty}\right)$
$\exists b \in \R: I = \left({-\infty \,.\,.\, b}\right)$


It follows by the definition of open real interval that $I$ has either an infimum $a$, or a supremum $b$, or both.

Thus the required value of $x$, from what has been proved above, is either $a$ or $b$.

$\blacksquare$


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