Prove that the balanced hull of a compact subset $K$ of a Hausdorff TVS $E$ is compact

Question

Let $E$ be a topological vector space Hausdorff (over $\mathbb{C}$) and $K \subset E$ compact. I want to prove that: if $M$ is the balanced hull, that is, $M$ is smallest balanced set containing $K$, then $M$ is compact.

But I didn't have any idea how to proceed.

Remembering that: a subset $A$ of a vector space $E$ is said to be balanced if for every $x\in A $ and every $\lambda \in \mathbb{C}$, $|\lambda|\leq 1$, we have $\lambda x \in A$.

Answer 1

Hint: Write $D$ for the set of complex numbers $z$ with $\lvert z\rvert \leq 1$. Then $M=DK$.