Metric Induced by Norm is Metric

Theorem
Let $V$ be a normed vector space.

Let $\norm{\,\cdot\,}$ denote its norm.

Let $d$ be the metric induced by $\norm{\,\cdot\,}$.

Then $d$ is a metric.

Proof of $\text M 1$ and $\text M 4$
Let $x, y \in V$.

Then:
 * $\map d {x, y} = \norm {x - y} \ge 0$

and furthermore:

Proof of $\text M 2$
Let $x, y, z \in V$.

Then:

Proof of $\text M 3$
Let $x, y \in V$.

Then:

As $d$ satisfies the metric space axioms, it is a metric.