Convex Subset of Topological Vector Space containing Zero Vector in Interior is Absorbing Set

Theorem

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

Let $\struct {X, \tau}$ be a topological vector space over $\GF$.

Let $C \subseteq X$ be a convex set such that ${\mathbf 0}_X \in C^\circ$, where $C^\circ$ denotes the interior of $C$.

Then $C$ is an absorbing set.

Proof

Let $x \in X$.

Let $V$ be an open neighborhood of ${\mathbf 0}_X$ contained in $C$.

Then from Multiple of Vector in Topological Vector Space Converges, we have:

$\ds \frac x n \to {\mathbf 0}_X$

So by the definition of a convergent sequence, for some $N \in \N$ we have:

$\ds \frac x N \in V$

Then:

$\ds x \in N V \subseteq N C$

Since $x \in X$ was arbitrary, we can apply Characterization of Convex Absorbing Set in Vector Space and obtain that:

$C$ is absorbing set.

$\blacksquare$