Real Number Line is Metric Space

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Theorem

Let $\R$ be the set of all real numbers.

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 $M1$

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

So axiom $M1$ holds for $d$.

$\Box$


Proof of $M2$

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

So axiom $M2$ holds for $d$.

$\Box$


Proof of $M3$

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

So axiom $M3$ holds for $d$.

$\Box$


Proof of $M4$

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

So axiom $M4$ holds for $d$.

$\blacksquare$


Sources