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:

$d \left({x_1, x_2}\right) = \left\vert{x_1 - x_2}\right\vert$

where $\left|{x}\right|$ is the absolute value of $x$.


Then $d$ is a metric on $\R$ and so $\left({\R, d}\right)$ is a metric space.


Proof

Proof of $M1$

\(\displaystyle d \left({x, x}\right)\) \(=\) \(\displaystyle \left\vert{x - x}\right\vert\) Definition of $d$
\(\displaystyle \) \(=\) \(\displaystyle 0\) Definition of Absolute Value

So axiom $M1$ holds for $d$.

$\Box$


Proof of $M2$

\(\displaystyle d \left({x, y}\right) + d \left({y, z}\right)\) \(=\) \(\displaystyle \left\vert{x - y}\right\vert + \left\vert{y - z}\right\vert\) Definition of $d$
\(\displaystyle \) \(\ge\) \(\displaystyle \left\vert{\left({x - y}\right) + \left({y - z}\right)}\right\vert\) Triangle Inequality for Real Numbers
\(\displaystyle \) \(=\) \(\displaystyle \left\vert{x - z}\right\vert\)
\(\displaystyle \) \(=\) \(\displaystyle d \left({x, z}\right)\) Definition of $d$

So axiom $M2$ holds for $d$.

$\Box$


Proof of $M3$

\(\displaystyle d \left({x, y}\right)\) \(=\) \(\displaystyle \left\vert{x - y}\right\vert\) Definition of $d$
\(\displaystyle \) \(=\) \(\displaystyle \left\vert{y - x}\right\vert\) Definition of Absolute Value
\(\displaystyle \) \(=\) \(\displaystyle d \left({y, x}\right)\) Definition of $d$

So axiom $M3$ holds for $d$.

$\Box$


Proof of $M4$

\(\displaystyle x\) \(\ne\) \(\displaystyle y\)
\(\displaystyle \implies \ \ \) \(\displaystyle \left\vert{x - y}\right\vert\) \(>\) \(\displaystyle 0\) Definition of Absolute Value
\(\displaystyle \implies \ \ \) \(\displaystyle d \left({x, y}\right)\) \(>\) \(\displaystyle 0\) Definition of $d$

So axiom $M4$ holds for $d$.

$\blacksquare$


Sources