Suppose that G is a region and let $\displaystyle a \in G$. Prove that G\{a} is a region.

So far I have divided the proof up into 2 parts, showing that G\{a} is open and showing that it is connected. I have done the open part, but am stuck on the connectedness.

I have assumed for contradiction that G\{a} is disconnected (generally seems to work for this type of question given the definition of connectedness), but cannot see where to get the contradiction from. Help much appreciated.