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:

$\mathbf 0_X \in A$

where $\mathbf 0_X$ denotes the zero vector in $X$.

Proof

From the definition of an absorbing subset, there exists $t \in \R_{>0}$ such that ${\mathbf 0}_X \in t A$.

So $\mathbf 0_X \in A$.

$\blacksquare$