Absorbing Set in Vector Space contains Zero Vector

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

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

Let $A \subseteq X$ be an absorbing subset of $X$.

Then ${\mathbf 0}_X \in A$.

Proof
From the definition of an absorbing subset, there exists $t > 0$ such that $0 \in t A$.

So ${\mathbf 0}_X \in A$.