The book I am reading says the following:
For any two variables $X$ and $Y$, if for every two sets $A$ and $B$ of real numbers $Pr(X \in A \cap Y \in B) =Pr(X \in A )Pr( Y \in B)$, then $X$ and $Y$ are independent. Further, if we can show that $P(X \leq x \cap Y \leq y) = P(X \leq x)P(Y \leq y)$, then we can conclude that $X$ and $Y$ are independent random variables.
Does anybody have a proof for this last statement? My book has omitted the proof.