Understanding the condition for independence of two random variables (Sheldon Ross) I don't understand the last line in the following paragraph.

How does one get that from the equation 4.3.7 using the 3 axioms of probability, which are mentioned in the book as follows:  

 A: Technically, that definition of independence is incorrect. Specifically the part:

for any [my emphasis] two sets of real numbers $A$ and $B$...

Because in general one cannot consider $P[X \in A]$ for any real subset $A$. The book even adds a footnote in the first page of the chapter on continuous random variables about this. 
For technical reasons one only works with sets $A\subset \mathbb{R}$ that are Borel sets. This means that the quoted part should read:

for any two real Borel sets $A$ and $B$...

This is important to your problem because it turns out that every Borel set can be build starting out with only sets of the form $\{r \in \mathbb{R}:r\leq c\}$, $c\in\mathbb{R}$ and applying finitely many times the operations of complement and countable unions. Actually, that characterizes the Borel sets of $\mathbb{R}$.
So, to show the equivalence of both conditions first note that the second condition is a special case of condition (4.3.7). The key idea for the other direction is: Since the sets $A$ and $B$ are Borel sets they can be written in therms of sets of the form $\{r \in \mathbb{R}:r\leq c\}$, $c\in\mathbb{R}$ as described above. Using the axioms, the probability operator can be distributed over that expression, you then can apply the weaker condition, and finally combine the probabilities again. To do this, consider the intermediate condition: For any Borel set $A\subset\mathbb{R}$, $b\in \mathbb{R} $: $$P\{X\in A, Y \leq b\} = P\{X\in A\}P\{Y \leq b\}.$$
Then go from your second condition to this intermediate condition and from there to (4.3.7).
A: First, Ross's condition (4.3.7) is overly broad -- $A$ and $B$ need to be restricted to be any two measurable sets of real numbers, not arbitrary sets of real numbers -- but is acceptable at the level of discourse of the book where the reader is not expected to know the difference. 
Next, the last sentence in the snippet glosses over the length of the proof that the condition in the definition is equivalent to $P(X\leq a, Y\leq b)  = P(X\leq a)P(Y\leq b)$ for all real numbers $a$ and $b$.  What is easy to show is that if $X$ and $Y$ are independent as per (4.3.7), then it follows that $P(X\leq a, Y\leq b)  = P(X\leq a)P(Y\leq b)$: we simply set $A= (-\infty,a]$ and $B=(-\infty, b]$ and apply (4.3.7).  (The fastidious should note that $A$ and $B$ are measurable sets of real numbers and no corners are being cut here).  The reverse implication gets us into measure theory and the like.
