P-Product Metric on Real Vector Space is Metric

Theorem
The $p$-product metric on the real vector space is a metric.

Comment on notation
It can be shown that:
 * $\displaystyle d_\infty \left({x, y}\right) = \lim_{p \to \infty} d_p \left({x, y}\right)$

That is:
 * $\displaystyle \lim_{p \to \infty} \left({\sum_{i \mathop = 1}^n \left \vert {x_i - y_i} \right \vert^p}\right)^{\frac 1 p} = \max_{i \mathop = 1}^n \left\{{\left \vert {x_i - y_i} \right \vert}\right\}$

Hence the notation.