# Newb question about limits and probability distribution functions

Background: I'm a newbie trying to learn about joint distribution functions. This is from Sheldon Ross's book (which is fantastic).

Question A: Why is it not kosher/acceptable to go from step 2 to step 6? [see below in red]

Question B: If someone went from step 2 to step 4, would this be seen as a totally newb maneuver? What is the logic behind these steps?

The transition from Step $3$ to Step $4$ is based on the Third Axiom of Probability which is generally stated in terms of countable unions of events but has an equivalent formulation:
This principle is used to go from the $\displaystyle P\big(\lim_{b\to \infty} (X \leq a, Y \leq b)\big)$ of Step $3$ to the $\displaystyle \lim_{b\to\infty} P(X\leq a, Y \leq b)$ of Step $4$.
Similarly, the $F(a,\infty)$ of Step $6$ is pretty meaningless since $F(\cdot, \cdot)$ is a function of real variables and so you cannot just substitute $\infty$ for one of them. It is, of course, quite common to define $F(a,\infty)$ as the symbol used as shorthand for the more elaborate $\displaystyle \lim_{b\to\infty} F(a,b)$ of Step $5$. Thus, jumping directly from Step $2$ directly to Step $6$ is not quite kosher (as you call it) until you have defined some notation at the very least. Ross is being very careful to cross the i's and dot the t's in his development of the material; just as well in light of the recent controversy over on stats.SE over one of his proofs in Chapter $2$ (where also Ross is perfectly correct in what he says).