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 and
Let $x, y \in V$.

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

and furthermore:

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

Then:

Proof of
Let $x, y \in V$.

Then:

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