Intersection of Balanced Sets in Vector Space is Balanced

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

Let $X$ be a vector space over $\Bbb F$.

Let $\sequence {E_\alpha}_{\alpha \mathop \in I}$ be an $I$-indexed family of balanced subsets of $X$.

Then:


 * $\ds E = \bigcap_{\alpha \mathop \in I} E_\alpha$ is balanced.

Proof
Let $x \in E$.

Then $x \in E_\alpha$ for all $\alpha \in I$.

Let $\lambda \in \Bbb F$ have $\cmod \lambda \le 1$.

Since $E_\alpha$ is balanced for each $\alpha \in I$, we have $\lambda x \in E_\alpha$ for each $\alpha \in I$.

So $\lambda x \in E$.