Non-Zero Dilation of Absorbing Set is Absorbing

Theorem

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

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

Let $A \subseteq X$ be an absorbing set.

Let $\lambda \in \GF \setminus \set 0$.

Then $\lambda A$ is absorbing.

Proof

Let $x \in X$.

Then there exists $t \in \R_{> 0}$ such that:

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

That is:

$\ds x \in \frac \alpha \lambda \paren {\lambda A}$ for $\cmod \alpha \ge t$.

Since the map $\alpha \mapsto \alpha/\lambda$ is a bijection from $\cmod \alpha \ge t$ to $\cmod \alpha \ge t/\cmod \lambda$, we have:

$\ds x \in \beta \paren {\lambda A}$ for $\cmod \beta \ge \dfrac t {\cmod \lambda}$.

So $\lambda A$ is absorbing.

$\blacksquare$