Euclidean Metric is Metric/Proof 1

Theorem

Let $M_{1'} = \struct {A_{1'}, d_{1'} }, M_{2'} = \struct {A_{2'}, d_{2'} }, \ldots, M_{n'} = \struct {A_{n'}, d_{n'} }$ be metric spaces.

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

The Euclidean metric on $\AA$ is a metric.

Proof

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

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

$\blacksquare$