collection of all entris of elements in compact subset in $M_n(\Bbb{C})$ is compact in $\Bbb{C}$

Question

Let $K_0$ be a compact subset of the space $M_n(\Bbb{C})$ and denote by $K\subseteq \mathbb{C}$ the collection of all matrix entries of elements in $K_0$. I can not see why $K$ is compact.

Thank you in advance!

Answer 1

The map $f_{ij} (A) = [A]_{ij}$ is continuous hence $f_{ij}(K_0) \subset \mathbb{C}$ is compact.

The union of a finite number of compact sets is compact, hence $K=\cup_{ij} f_{ij}(K_0)$ is compact.