The existence of a countable dense subset

Question

Suppose that $E$ is a normed vector space of finite dimension. Suppose $A$ and $B$ are nonempty convex subsets of $E$ satisfying $A\cap B=\emptyset$. Let $C$ be $\{x-y|x\in A,y\in B\}$. Show that $C$ admits a countable dense subset.

Answer 1

If $E$ is $\mathbb{R}^n$ or $\mathbb{C}^n$:

Every non empty subset $C$ of $\mathbb{R}^n$ or $\mathbb{C}^n$ admits a countable dense subset. Take $F=\mathbb{Q}^n$ or $F=(\mathbb{Q}+i \mathbb{Q})^n$. $F$ is countable. For every $f \in F$ and $k \in \mathbb{N}^*$, $\exists c_{f,k} \in C$ such that $d(f,c_{f,k}) <d(f,C)+\frac{1}{k}$.

Then $D=\{ c_{f,k} | f \in F, k \in \mathbb{N}^*\}$ is a countable dense subset in $C$.