Subset of von Neumann-Bounded Set is von Neumann-Bounded

Definition

Let $\Bbb F \in \set {\R, \C}$.

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

Let $E \subseteq X$ be von Neumann-bounded.

Let $F \subseteq X$.

Then $F$ is von Neumann-bounded.

Proof

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

Then there exists $s > 0$ such that:

$E \subseteq t V$ for each $t > s$.

Then:

$F \subseteq t V$ for each $t > s$.

$\blacksquare$