Let A, B and C be subsets of a topological space X with C⊂A ∪ B. If A, B and A ∪ B are given the relative topologies, prove that C is open with respect to A ∪ B if and only if C∩A is open with respect to A and C∩B is open with respect to B.

I need help with the part "if C∩A is open with respect to A and C∩B is open with respect to B then C is open with respect to A ∪ B"

I already did the other direction. This is what I gathered, but have not had much luck connecting everything.

We know that since C∩A is open with respect to A then C∩A=G∩A for some open set G on the topological space. Similarly, since C∩B is open with respect to B, then C∩B=H∩B for some open set H on the topological space. I tried using G∪H as my open set, and I cant seem to prove that (G∪H)∩(A ∪ B)=C, but to be honest I have tried to come up with something else, and have ran out of luck. How do I prove the existence of an open set N such that N∩(A ∪ B)=C?

Thanks for any help in advance

(This exercise is taken from Schaum's Outlines of General topology chapter 5 problem 90)