I have written a valid proof that f(A) is bounded. I don't know how to show that f(B) is also closed.
I am wondering if I can use the Completeness Theorem to argue that there is a supremum and infimum. Then I would need to show that those two numbers reside in A for A to be closed and not open. I don't know how to do that, but am I at least on the right track?
Nevermind, I found out that I am on the right track, and that I just need to use the Maximum-Minimum Theorem.