# Real Number Line is Metric Space

## 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 $\text M 2$

 $\ds \map d {x, y} + \map d {y, z}$ $=$ $\ds \size {x - y} + \size {y - z}$ 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 $\text M 2$ 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 $d$.

$\blacksquare$