Suppose $X,Y \subset \mathbb{R}$ and let $f(x,y): X\times Y \rightarrow \mathbb{R}$ be non-decreasing in both arguments. Further, suppose that $x_{2}\geq x_{1}$ and $y_{2}\geq y_{1}$. I am trying to prove (though I am not entirely sure it holds) that the above assumptions imply that $f(x_{2}, y_{2}) - f(x_{1}, y_{2}) \geq f(x_{2}, y_{1}) - f(x_{1}, y_{1})$.
I have first tried to come up with counterexamples, but have not found any yet. Then I have tried to prove it directly, that is by deriving it directly from the assumptions, but I have not succeeded. I have also tried to show that the contrapositive statement will yield a contradiction, but I have only been successful in showing that it yields contradictions in specific counterexamples, and not in the general case. As an example, I have been able to show that if $f(x_{2}, y_{2}) - f(x_{1}, y_{2}) < f(x_{2}, y_{1}) - f(x_{1}, y_{1})$, then letting $f(x,y)=x+y$ and setting $y_{1}=y_{2} = 0$ yields a contradiction, namely that $x_{2} - x_{1} < x_{2} - x_{1}$. Hence, this inequality cannot hold in general. I still have not been able to show that it leads to a contradiction in the general case.
Any help on proving/disproving in the statement would be much appreciated. The reason I want to prove/disprove the statement is that if it turns out to be true, it will help me understand the proof of a lemma (lemma 2.1.5) in the book An Introduction to Copulas by Roger B. Nelsen.
Thanks in advance to all that took the time to read this post.