Superset of Absorbing Set is Absorbing

Theorem

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

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

Let $A$ be an absorbing set.

Let $B \supseteq A$.

Then $B$ is absorbing.

Proof

Let $x \in X$.

Since $A$ is absorbing, there exists $t \in \R_{> 0}$ such that:

$x \in t A$ for $\cmod \alpha \ge t$.

Since $A \subseteq B$, we obtain:

$x \in t B$ for $\cmod \alpha \ge t$.

$\blacksquare$