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
Checking the axioms M1', M2, M3 and M4' for a metric:

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

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

M2
Let $x, y, z \in V$. Then:

M3
Let $x, y \in V$. Then:

As $d$ satisfies the four axioms M1', M2, M3 and M4', it is a metric.