Metric Induced by Norm is Invariant Metric

Theorem
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.

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

Then $d$ is translation-invariant.

Proof
Let $x, y, z \in X$.

Then, we have: