Helping with proofs involving connected sets is very difficult without knowing the exact definition and sequence of theorems in your textbook. There are many different but equivalent approaches to connected sets. This particular question is either a simple consequence of the theorem-“The closure of a connected set is connected”.Or it may be a lemma to the proof of that theorem. Basically you would show that if Au{p} is separated some open set contains p and no other point of A. That is a contradiction.