Proof of: If $P(A) = P(B) = 1$ then $P(A \cap B) = 1$.

Well, you cannot say $P(A\cap B)= P(A) \cdot P(B)$ because you do not know if $A$ and $B$ are independent. Anyway, you could say $$P(A\cap B) = 1-P((A\cap B)^c)=1-P(A^c \cup B^c) \geq 1-(P(A^c)+ P(B^c))= 1-(0+0)=1$$

We begin with $$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$ Since $P(A)=P(B)=1$, we can rearrange this as $$P(A\cap B)=2-P(A\cup B)$$ While we might not know $P(A\cup B)$, we know that $P(A\cup B)\le 1$ (true for all probabilities), and hence $$P(A\cap B)\ge 2-1=1$$ Since also $P(A\cap B)\le 1$ (true for all probabilities), we can be sure that $P(A\cap B)=1$.

Note that $A\subset A\cup B$ and $B\subset A\cup B$ implies $P(A\cup B)\geq P(A)$ and $P(A\cup B)\geq P(B)$. Then $P(A\cup B)=1$. By Inclusion–exclusion principle $$ P(A\cup B)= P(A)+P(B)-P(A\cap B). $$ So $$ 1=1+1-P(A\cap B). $$