If $A\cap B^\complement=\emptyset$ then $A\cap B=A$ I need help with this excercise.
If $A\cap B^\complement=\emptyset$ then $A\cap B=A$

I try
$$A\cap B^\complement =\emptyset$$
$$(A\cap B^\complement)\cap B =\emptyset \cap B$$
$$A\cap (B^\complement\cap B )= B$$
$$A=B$$
Is this reasoning correct?

No, your reasoning leads nowhere because you get $\emptyset=\emptyset$. The property you want is a direct consequence of
$$
A=(A\cap B)\cup(A\cap B^c). 
$$
