Taxicab Metric is Metric

Theorem
The taxicab metric is a metric.

Proof
The taxicab metric is as follows:

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

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

The taxicab metric on $$\reals^n$$ is $$d_1 \left({x, y}\right) = \sum_{i=1}^n \left|{x_i - y_i}\right|$$.

It is easy to see that conditions M0, M1 and M2 of the conditions for being a metric are satisfied. So all we need to do is check M3.

That is, that $$\sum_{i=1}^n \left|{x_i - z_i}\right| \le \sum_{i=1}^n \left|{x_i - y_i}\right| + \sum_{i=1}^n \left|{y_i - z_i}\right|$$.

Proof by induction:

For all $$n \in \mathbb{N}^*$$, let $$P \left({n}\right)$$ be the proposition that the taxicab metric is a metric on $$\reals^n$$.


 * $$P(1)$$ is true, as this just says $$\left|{x - z}\right| \le \left|{x - y}\right| + \left|{y - z}\right|$$.

This is the triangle inequality for real numbers.

Basis for the Induction

 * $$P(2)$$ is the following case:

$$ $$ $$ $$

This is our basis for the induction.

Induction Hypothesis

 * Now we need to show that, if $$P \left({k}\right)$$ is true, where $$k \ge 2$$, then it logically follows that $$P \left({k+1}\right)$$ is true.

So this is our induction hypothesis:

$$\sum_{i=1}^k \left|{x_i - z_i}\right| \le \sum_{i=1}^k \left|{x_i - y_i}\right| + \sum_{i=1}^k \left|{y_i - z_i}\right|$$.

Then we need to show:

$$\sum_{i=1}^{k+1} \left|{x_i - z_i}\right| \le \sum_{i=1}^{k+1} \left|{x_i - y_i}\right| + \sum_{i=1}^{k+1} \left|{y_i - z_i}\right|$$.

Induction Step
This is our induction step:

$$ $$ $$ $$ $$

So $$P \left({k}\right) \Longrightarrow P \left({k+1}\right)$$ and the result follows by the Principle of Mathematical Induction.

Therefore $$d_1 \left({x, y}\right)$$ is a metric on $$\reals^n$$ for all $$n$$.