# Euclidean Metric on Real Number Line is Metric

## Theorem

The Euclidean metric on the real number line $\R$ is a metric.

## Proof 1

The Euclidean metric on the real number line is a special case of the Euclidean metric on the real vector space $\R^n$.

The result follows from Euclidean Metric on Real Vector Space is Metric.

$\blacksquare$

## Proof 2

Consider the real number line under the Euclidean metric:

$M = \struct {\R, d}$

where $d$ is the distance function given by:

$\map d {x, y} = \size {x - y}$

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

 $\ds \map d {x, x}$ $=$ $\ds \size {x - x}$ Definition of $d$ $\ds$ $=$ $\ds \size 0$ $\ds$ $=$ $\ds 0$

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 $d$ $\ds$ $=$ $\ds \size {y - x}$ $\ds$ $=$ $\ds \map d {y, x}$ 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 \ \$ $\ds x - y$ $\ne$ $\ds 0$ $\ds \leadsto \ \$ $\ds \size {x - y}$ $>$ $\ds 0$ $\ds \leadsto \ \$ $\ds \map d {x, y}$ $>$ $\ds 0$ Definition of $d$

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

$\blacksquare$