Metric Induced by Norm is Metric

Theorem
Let $V$ be a normed vector space, and denote $\left\Vert{\cdot}\right\Vert$ for its norm.

Let $d$ be the metric induced by $\left\Vert{\cdot}\right\Vert$.

Then $d$ is a metric.

Proof of $M1$ and $M4$
Let $x, y \in V$.

Then $d \left({x, y}\right) = \left\Vert{x - y}\right\Vert \ge 0$, and furthermore:

Proof of $M2$
Let $x, y, z \in V$.

Then:

Proof of $M3$
Let $x, y \in V$.

Then:

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