# Real Number Line is Metric Space

## 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 $d$.

$\blacksquare$