Real Number Line is Metric Space

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Theorem

Let $\R$ be the real number line.

Let $d: \R \times \R \to \R$ be defined as:

$\map d {x_1, x_2} = \size {x_1 - x_2}$

where $\size x$ is the absolute value of $x$.


Then $d$ is a metric on $\R$ and so $\struct {\R, d}$ is a metric space.


Proof

Proof of Metric Space Axiom $(\text M 1)$

\(\ds \map d {x, x}\) \(=\) \(\ds \size {x - x}\) Definition of $d$
\(\ds \) \(=\) \(\ds 0\) Definition of Absolute Value

So Metric Space Axiom $(\text M 1)$ holds for $d$.

$\Box$


Proof of Metric Space Axiom $(\text M 2)$: Triangle Inequality

\(\ds \map d {x, y} + \map d {y, z}\) \(=\) \(\ds \size {x - y} + \size {y - z}\) Definition of $d$
\(\ds \) \(\ge\) \(\ds \size {\paren {x - y} + \paren {y - z} }\) Triangle Inequality for Real Numbers
\(\ds \) \(=\) \(\ds \size {x - z}\)
\(\ds \) \(=\) \(\ds \map d {x, z}\) Definition of $d$

So Metric Space Axiom $(\text M 2)$: Triangle Inequality holds for $d$.

$\Box$


Proof of Metric Space Axiom $(\text M 3)$

\(\ds \map d {x, y}\) \(=\) \(\ds \size {x - y}\) Definition of $d$
\(\ds \) \(=\) \(\ds \size {y - x}\) Definition of Absolute Value
\(\ds \) \(=\) \(\ds \map d {y, x}\) Definition of $d$

So Metric Space Axiom $(\text M 3)$ holds for $d$.

$\Box$


Proof of Metric Space Axiom $(\text M 4)$

\(\ds x\) \(\ne\) \(\ds y\)
\(\ds \leadsto \ \ \) \(\ds \size {x - y}\) \(>\) \(\ds 0\) Definition of Absolute Value
\(\ds \leadsto \ \ \) \(\ds \map d {x, y}\) \(>\) \(\ds 0\) Definition of $d$

So Metric Space Axiom $(\text M 4)$ holds for $d$.

$\blacksquare$


Sources

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