Subadditivity of Invariant Metric on Vector Space

Theorem
Let $K$ be a field.

Let $X$ be a vector space over $K$.

Let $d$ be an invariant metric on $X$.

Then:
 * $\map d {n x, 0} \le n \map d {x, 0}$

for each $n \in \N$ and $x \in X$.

Proof
We have: