P-Product Metric on Real Vector Space is Metric

Theorem
Let $\R^n$ be an $n$-dimensional real vector space.

Let $p \in \R_{\ge 1}$.

Let $d_p: \R^n \times \R^n \to \R$ be the $p$-product metric on $\R^n$:


 * $\ds \map {d_p} {x, y} := \paren {\sum_{i \mathop = 1}^n \size {x_i - y_i}^p}^{\frac 1 p}$

where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \R^n$.

Then $d_p$ is a metric.

Comment on notation
It can be shown that:
 * $\ds \map {d_\infty} {x, y} = \lim_{p \mathop \to \infty} \map {d_p} {x, y}$

That is:
 * $\ds \lim_{p \mathop \to \infty} \paren {\sum_{i \mathop = 1}^n \size {x_i - y_i}^p}^{\frac 1 p} = \max_{i \mathop = 1}^n \set {\size {x_i - y_i} }$

Hence the notation.