Norm on Vector Space is Seminorm

Theorem
Let $\GF \in \set {\R, \C}$.

Let $X$ be a vector space over $\GF$.

Let $\norm {\, \cdot \,}$ be a norm on $X$.

Then $\norm {\, \cdot \,}$ is a seminorm.

Proof
Note that and  are precisely  and.