For the univariate case, I understand how to create bracketing functions for the CDF, $F$. For example, see here. It is clear to me that there exists a partition $t_1,\cdots,t_k$ that lead to a cover. However, I cannot figure out how to create bracketing functions for the case of a bivariate distribution function. I have tried the following:
I believe there exist $(x_1, y_1),\ldots,(x_k,y_k)$ such that $x_1,y_1=-\infty,x_k,y_k=\infty$ and that if we define
$l_j=1\{x\le x_{j-1},y\le y_{j-1}\}$
$u_j=1\{x < x_j,y < y_j\}$
then $||l-u||<\epsilon$
But I don't see how they can cover all functions $1\{x\le a,y\le b\}$. That is, I don't see how (for a fixed $\epsilon$) we can construct the points $(x_j, y_j)$ above such that for every fixed a,b, we have that there exists j such that
$P(l_j(X,Y)\le 1\{X\le a,Y\le b\}\le u_j(X,Y))=1$
I feel if you give me a grid, I will always be able to find a a,b that do not satisfy the above for any j.
Am I right that it is impossible to create the covering functions as above? If so, can you provide bracketing functions that do work? Please do not prove that they work (I would like to attempt that first).