This took a bit of thought, but I figured an epsilon chase works best.

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: