Taxicab Metric on Real Vector Space is Metric/Proof 2

Theorem
The taxicab metric on the real vector space $\R^n$ is a metric.

Proof
The taxicab metric on $\R^n$ is:
 * $\displaystyle d_1 \left({x, y}\right) = \sum_{i \mathop = 1}^n \left\vert{x_i - y_i}\right\vert$

for $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \R^n$.

Proof of $M1$
So axiom $M1$ holds for $d_1$.

Proof of $M2$
So axiom $M2$ holds for $d_1$.

Proof of $M3$
So axiom $M3$ holds for $d_1$.

Proof of $M4$
So axiom $M4$ holds for $d_1$.