Let A and B be subsets of with , denoting the sets of interior points for A and B respectively. Prove that is a subset of the interior of AUB. Give an example where the inclusion is strict.
One reason we ask people to show there work is that there are a number of different ways of defining "interior points" and without seeing what you are trying we don't know which is appropriate for you. I suspect you are using "p is an interior point of p if and only if there exist a neighborhood of p that is a subset of A".
So if p is in then it is in both and . Since it is in there exist some neighborhood of p, ( is the radius), such that . Since p is in , there exist some neighborhood of p, such that .
Can you finish now?