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:

$\map d {n x, {\mathbf 0}_X} \le n \map d {x, {\mathbf 0}_X}$

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

We have:

$\ds \map d {n x, {\mathbf 0}_X}$

$=$

$\ds \sum_{k \mathop = 1}^n \map d {k x, \paren {k - 1} x}$

$\ds $

$\le$

$\ds \sum_{k \mathop = 1}^n \map d {k x - \paren {k - 1} x, \paren {k - 1} x - \paren {k - 1} x}$

$\ds $

$=$

$\ds \sum_{k \mathop = 1}^n \map d {x, {\mathbf 0}_X}$

$\ds $

$=$

$\ds n \map d {x, {\mathbf 0}_X}$

$\blacksquare$