Let $X_1$ and $X_2$ be independent random variables of the discrete type with joint p.m.f $\ p_1(x_1)\cdot p_2(x_2), \ (x_1,x_2)\in \mathscr{A}$. Let $y_1 = u_1(x_1)$ and $y_2=u_2(x_2)$ denote a one-to-one transformation that maps $\mathscr{A}$ onto $\mathscr{B}$. Show that $Y_1 = u_1(X_1)$ and $Y_2 = u_2(X_2)$ are independent.
For this exercise, I'm not supposed to use the c.d.f. technique. Rather, I'm supposed to the change of variable technique using the given statements in the exercise. Also, $\mathscr{B}=\{(y_1,y_2):y_1 = u_1(x_1) \text{ and }y_2 = u_2(x_2)\}$. I'll outline what I've gotten to so far, and where I got stuck.
My approach to show that $Y_1=u_1(X_1)$ and $Y_2=u_2(X_2)$ are independent is to show that the joint p.m.f. of $Y_1$ and $Y_2$, $g(y_1,y_2)=g(y_1)\cdot g(y_2)$.
Since the transformation is bijective, we know the inverses $X_1=u_1^{-1}(Y_1)$ and $X_2=u_2^{-1}(Y_2)$ exist.
This is as far as I got. Please help. I know this was answered here but I think it's using the c.d.f. technique (I think): https://math.stackexchange.com/questions/209176/probability-and-random-variable-with-joint-pdf