basic doubt about connectness - general topology

basic doubt about connectness - general topology

Let $E \subset R^n$, where $E^c$ is disconnected.
Then exists $U,V \subset R^n$, $U,V \neq \emptyset$ disjoint, and open
relative to $E^c$.
We have too $U \cap \overline{V} = \emptyset = V \cap \overline{U} $ . My
book says :
Exists a ball $B \subset R^n$ such that $B$ intersects both
${\overline{U}}^c$ and ${\overline{V}}^c$.
Drawing a picture is easy to see this last affirmation, but i dont know
how to prove that. Someone can give a hint ? Any help is apreciated.
Thanks in advance!