I have to prove that
if $A \subseteq \mathbb{R}$ is countable, then $\exists x \in \mathbb{R}\, (x+A) \cap A = \emptyset, $
where $x+A$ denotes the set $\{x + a \mid a \in A\}$.
I can see why this is true for some specific subsets (like the set of rationals or the set of algebraic numbers), but the general approach eludes me. Any hints would be appreciated.