$\bigcup_{\alpha\in C}\alpha A$ is compact in the topological vector space $X$

Question

Let $X$ be a topological vector space, with $A \subset X$ and $C \subset \mathbb{K}$ both compact subsets.

Why is $\bigcup_{\alpha\in C}\alpha A$ compact in $X$?

Answer 1

$\cup_{\alpha \in C} \alpha A$ is the image of $C\times A$ under the continuous map $(c,y) \to cy$.

Answer 2

Hint: $C\times A$ is compact in $\mathbb{K}\times X$ and multiplication by scalars is continuous.