Of course we are assuming that A and B are independent events. I know how to show that if P(A)=1 then P(B)=P(AB), but how do we show that if P(A)=0?
Kyle, from your title, it seems you are asking, if $P(A) = 0$, how can we prove that $A$ and $B$ are independent events? The condition that must hold for two events, $A$ and $B$, to be independent is
$$P(AB) = P(A)P(B)$$
So, if you want to prove $A$ and $B$ are independent, you need to show this. In this case, if $P(A) = 0$, what is the right hand side? And, since $P(AB)$ means the probability of $A$ and $B$ both happening, what do you think $P(AB)$ is when the probability of just $A$ happening is $0$?
Note, in your question, you actually ask something totally different. You assume $A$ and $B$ are independent and you want to prove $P(B) = P(AB)$. This is a vastly different question.