Euclidean Metric is Metric/Proof 1

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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

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

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

$\blacksquare$