$\overline {\text {span}_{\mathbb {Q}}A}=\text {span}_{\mathbb {R}}B$

Question

Let $(X,\|.\|)$ be a Banach space. Let $A$ and $B$ are subset of $X$ such that $B$ is countable set.

Show that :

$$ \overline {A}=B\Rightarrow \overline {\text {span}_{\mathbb {Q}}A}=\text {span}_{\mathbb {R}}B $$

An idea please.