Taxicab Metric on Metric Space Product is Continuous

Theorem
Let $M = \struct {A, d}$ be a metric space.

Let $\AA$ be the Cartesian product $A \times A$.

Let $d_1$ be the taxicab metric on $\AA$:
 * $\ds \map {d_1} {x, y} = \sum_{i \mathop = 1}^n \map {d_{i'} } {x_{i'}, y_{i'} }$

for $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \AA$.

Then $d_1: \AA \to \AA$ is a continuous function.

Proof
A direct application of Metric is Continous Mapping.