we are asked to prove that $X$ and $Y$ are independent if and only if $\{a< X\le b\}$ and $\{c<Y\le d\}$ are independent sets. However we only have that $\{X\le b\}$ and $\{Y\le d\}$ are independent?
Do we work with compliments of $\{X>a\}$ and $\{Y>c\}$? and then the union of those sets are somehow independent?
Hint:
$$P(a<X\leq b\wedge c<Y\leq d)=P(X\leq b\wedge c<Y\leq d)-P(X\leq a\wedge c<Y\leq d)=$$$$\left[P(X\leq b\wedge Y\leq d)-P(X\leq b\wedge Y\leq c)\right]-\left[P(X\leq a\wedge Y\leq d)-P(X\leq a\wedge Y\leq c)\right]$$