Let and be given. Then is in which is open and so there exists an such that .
Now for the trickier bits, I'll give a few helpful nudges. What we want to show is that there exists a delta such that whenever . We already found an epsilon neighborhood of , so we want to construct some points in whose images lie in this neighborhood. This is how we want to go about picking our delta.
Big idea 1:Spoiler:
Big idea 2:Spoiler:
Big idea 3:Spoiler: