Euclidean Metric on Real Vector Space is Metric/Proof 2

Theorem

The Euclidean metric on a real vector space $\R^n$ is a metric.

Proof

Consider the Euclidean space $M = \struct {\R^n, d_2}$ where $d_2$ is the distance function given by:

$\ds \map {d_2} {x, y} = \paren {\sum_{i \mathop = 1}^n \paren {x_i - y_i}^2}^{\frac 1 2}$

where $x = \tuple {x_1, x_2, \ldots, x_n}$ and $y = \tuple {y_1, y_2, \ldots, y_n}$.

Proof of Metric Space Axiom $\text M 1$

 $\ds \map {d_2} {x, x}$ $=$ $\ds \paren {\sum_{i \mathop = 1}^n \paren {x_i - x_i}^2}^{\frac 1 2}$ Definition of $d_2$ $\ds$ $=$ $\ds \paren {\sum_{i \mathop = 1}^n 0^2}^{\frac 1 2}$ $\ds$ $=$ $\ds 0$

So Metric Space Axiom $\text M 1$ holds for $d_2$.

$\Box$

Metric Space Axiom $\text M 2$

It is required to be shown:

$\map {d_2} {x, y} + \map {d_2} {y, z} \ge \map {d_2} {x, z}$

for all $x, y, z \in \R^n$.

Let:

$(1): \quad z = \tuple {z_1, z_2, \ldots, z_n}$
$(2): \quad$ all summations be over $i = 1, 2, \ldots, n$
$(3): \quad x_i - y_i = r_i$
$(4): \quad y_i - z_i = s_i$.

Thus we need to show that:

$\ds \paren {\sum \paren {x_i - y_i}^2}^{\frac 1 2} + \paren {\sum \paren {y_i - z_i}^2}^{\frac 1 2} \ge \paren {\sum \paren {x_i - z_i}^2}^{\frac 1 2}$

We have:

 $\ds \map {d_2} {x, y} + \map {d_2} {y, z}$ $=$ $\ds \paren {\sum \paren {x_i - y_i}^2}^{\frac 1 2} + \paren {\sum \paren {y_i - z_i}^2}^{\frac 1 2}$ Definition of $d_2$ $\ds$ $=$ $\ds \paren {\sum r_i^2}^{\frac 1 2} + \paren {\sum s_i^2}^{\frac 1 2}$ $\ds$ $\ge$ $\ds \paren {\sum \paren {r_i + s_i}^2}^{\frac 1 2}$ Minkowski's Inequality for Sums: index $2$ $\ds$ $=$ $\ds \paren {\sum \paren {x_i - y_i + y_i - z_i}^2}^{\frac 1 2}$ Definition of $r_i$ and $s_i$ $\ds$ $=$ $\ds \paren {\sum \paren {x_i - z_i}^2}^{\frac 1 2}$ $\ds$ $=$ $\ds \map {d_2} {x, z}$ Definition of $d_2$

So Metric Space Axiom $\text M 2$ holds for $d_2$.

$\Box$

Metric Space Axiom $\text M 3$

 $\ds \map {d_2} {x, y}$ $=$ $\ds \paren {\sum \paren {x_i - y_i}^2}^{\frac 1 2}$ Definition of $d_2$ $\ds$ $=$ $\ds \paren {\sum \paren {y_i - x_i}^2}^{\frac 1 2}$ $\ds$ $=$ $\ds \map {d_2} {y, x}$ Definition of $d_2$

So Metric Space Axiom $\text M 3$ holds for $d_2$.

$\Box$

Metric Space Axiom $\text M 4$

 $\ds x$ $\ne$ $\ds y$ $\ds \leadsto \ \$ $\ds \exists k \in \closedint 1 n: \,$ $\ds x_k$ $\ne$ $\ds y_k$ $\ds \leadsto \ \$ $\ds x_k - y_k$ $\ne$ $\ds 0$ $\ds \leadsto \ \$ $\ds \paren {\sum \paren {x_k - y_k}^2}^{\frac 1 2}$ $>$ $\ds 0$ $\ds \leadsto \ \$ $\ds \map {d_2} {x, y}$ $>$ $\ds 0$ Definition of $d_2$

So Metric Space Axiom $\text M 4$ holds for $d_2$.

$\blacksquare$