I have two variables $X$ and $Y$ with the following joint probablity density function
$ f (x,y) = \begin{cases} \frac14 (1+xy) & \text{if } |x| < 1, |y| < 1\\ &\\ 0 & \text{otherwise} \end{cases} $
The problem is to prove that $X$ and $Y$ are not independent, but that $X^2$ and $Y^2$ are. I calculated the marginal density functions of both $X$ and $Y$ and since their product doesn't equal the marginal density function, I proved they are not independent.
However, I wasn't sure about the second part and reviewed the solution. In the given solution, independence was not proven by the following.
$ f_{x^2,y^2} (u,v) = f_{x^2} (u) \centerdot f_{y^2} (v) $
Instead, it was proven that the cumulative distribution functions exhibit this property and that this implies independence.
$ P (X^2 \leq u \cap Y^2 \leq v) = P (X^2 \leq u) \centerdot P (Y^2 \leq v) $
I couldn't find any reference that said this implied independence. I believed such implication only worked with the density functions. This is also not mentioned on the wikipedia page for cumulative distribution function.
So, I'm wondering, is this also a way to prove independence? Can I use this technique when possible?