Borel measure - unbounded sets

Question

Let $E\subset \mathbb{R}$ be borel-measurable and $\mu (E) < + \infty $, I have to show the following: $$ \lim_{x \to +\infty} \mu ((E + \{ x \}) \cap E)=0 $$ A hint suggests to first consider bounded sets $E$.
My proof for bounded sets would be:
$E$ bounded $\Rightarrow \exists L \in \mathbb{R} $, s.t. $\forall a \in E: |a|< L \Rightarrow \forall x > 2L: x > $ sup$(E) - $ inf$(E)$
$ \Rightarrow $ inf$(E) + x > $ sup $ (E) \Rightarrow \forall x > 2L: \mu ((E + \{ x \}) \cap E)= \mu( \emptyset )= 0$
$ \Rightarrow \lim_{x \to +\infty} \mu ((E + \{ x \}) \cap E)=0 $

However, I do not really have an idea how to use that to show it for $E$ unbounded, because $E$ unbounded and $\mu (E) < + \infty$ seems counterintuative to me.