Suppose that E is a connected proper subset of R^n and that E is a proper subset of A which is a proper subset of the closure of E. Prove that A is connected.
I can't seem to really find a starting point. My general approach to showing a set is connected is to either reach a contradiction assuming it is disconnected or to show that the set is convex (and thus connected).
Hints are greatly appreciated!
Our definition of connectedness is: A set E is connected if and only if it cannot be separated by any pair of relatively open sets. Otherwise, it is disconnected.