# Euclidean Metric is Metric

## Theorem

Let $M_{1'} = \left({A_{1'}, d_{1'}}\right), M_{2'} = \left({A_{2'}, d_{2'}}\right), \ldots, M_{n'} = \left({A_{n'}, d_{n'}}\right)$ be metric spaces.

Let $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_{i'}$ be the cartesian product of $A_{1'}, A_{2'}, \ldots, A_{n'}$.

The Euclidean metric on $\mathcal A$ is a metric.

## Proof 1

The Euclidean metric on $\mathcal A$ is a special case of the $p$-product metric.

The result follows from $p$-Product Metric is Metric.

$\blacksquare$

## Proof 2

We have that the Euclidean metric on $\mathcal A$ is defined as:

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

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

### Proof of $M1$

 $\displaystyle \map {d_2} {x, x}$ $=$ $\displaystyle \paren {\sum_{i \mathop = 1}^n \paren {\map {d_{i'} } {x_i, x_i} }^2}^{\frac 1 2}$ Definition of $d_2$ $\displaystyle$ $=$ $\displaystyle \paren {\sum_{i \mathop = 1}^n 0^2}^{\frac 1 2}$ as $d_{i'}$ fulfils axiom $M1$ $\displaystyle$ $=$ $\displaystyle 0$

So axiom $M1$ holds for $d_2$.

$\Box$

### Proof of $M2$

Let:

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

Thus we need to show that:

$\displaystyle \paren {\sum \paren {\map {d_{i'} } {x_i, y_i} }^2}^{\frac 1 2} + \paren {\sum \paren {\map {d_{i'} } {y_i, z_i} }^2}^{\frac 1 2} \ge \paren {\sum \paren {\map {d_{i'} } {x_i, z_i} }^2}^{\frac 1 2}$

We have:

 $\displaystyle \map {d_2} {x, y} + \map {d_2} {y, z}$ $=$ $\displaystyle \paren {\sum \paren {\map {d_{i'} } {x_i, y_i} }^2}^{\frac 1 2} + \paren {\sum \paren {\map {d_{i'} } {y_i, z_i} }^2}^{\frac 1 2}$ Definition of $d_2$ $\displaystyle$ $=$ $\displaystyle \paren {\sum r_i^2}^{\frac 1 2} + \paren {\sum s_i^2}^{\frac 1 2}$ $\displaystyle$ $\ge$ $\displaystyle \paren {\sum \paren {r_i + s_i}^2}^{\frac 1 2}$ Minkowski's Inequality for Sums: index $2$ $\displaystyle$ $=$ $\displaystyle \paren {\sum \paren {\map {d_{i'} } {x_i, y_i} + \map {d_{i'} } {y_i, z_i} }^2}^{\frac 1 2}$ Definition of $r_i$ and $s_i$ $\displaystyle$ $\ge$ $\displaystyle \paren {\sum \paren {\map {d_{i'} } {x_i, z_i} }^2}^{\frac 1 2}$ as $d_{i'}$ fulfils axiom $M2$ $\displaystyle$ $=$ $\displaystyle \map {d_2} {x, z}$ Definition of $d_2$

So axiom $M2$ holds for $d_2$.

$\Box$

### Proof of $M3$

 $\displaystyle \map {d_2} {x, y}$ $=$ $\displaystyle \paren {\sum \paren {\map {d_{i'} } {x_i, y_i} }^2}^{\frac 1 2}$ Definition of $d_2$ $\displaystyle$ $=$ $\displaystyle \paren {\sum \paren {\map {d_{i'} } {y_i, x_i} }^2}^{\frac 1 2}$ as $d_i$ fulfils axiom $M3$ $\displaystyle$ $=$ $\displaystyle \map {d_2} {y, x}$ Definition of $d_2$

So axiom $M3$ holds for $d_2$.

$\Box$

### Proof of $M4$

 $\displaystyle x$ $\ne$ $\displaystyle y$ $\displaystyle \leadsto \ \$ $\displaystyle \exists k \in \closedint 1 n: x_k$ $\ne$ $\displaystyle y_k$ $\displaystyle \leadsto \ \$ $\displaystyle \map {d_k} {x_k, y_k}$ $>$ $\displaystyle 0$ as $d_k$ fulfils axiom $M4$ $\displaystyle \leadsto \ \$ $\displaystyle \paren {\sum \paren {\map {d_{i'} } {x_i, y_i} }^2}^{\frac 1 2}$ $>$ $\displaystyle 0$ $\displaystyle \leadsto \ \$ $\displaystyle \map {d_2} {x, y}$ $>$ $\displaystyle 0$ Definition of $d_2$

So axiom $M4$ holds for $d_2$.

$\blacksquare$