A probability function is determined on a dense set- Where is density used in the following proof? A probability function is determined on a dense set- Where is density used in the following proof?
Consider the following theorem and proof from Resnick's book A probability path. I cannot really see where the assumption that $D $ is dense in $R $ is used. Can you enlighten me? Is it needed for $(8.2) $? I think there would be sufficient if there existed an $x' \in D$ such that $x' \ge x $ for it to hold.


Thanks in advance!

The Lemma proves two things at the same time: I) a distribution function is uniquely determined by its restriction to a dense set, and II) a function $F_D$ defined on a dense set and having the listed properties extends to a distribution function.  Part (I) is the more important part, I suppose.
For (II) you are right that denseness of $D$ is not necessary; it is sufficient that $D$ has arbitrarily large elements.  Note that the second equality in (8.1) may not hold if $D$ is not dense, because you may not be able to let $y$ tend to $x$ from above within $D$.
The proof of (I) is by noting that any distribution function $F$ must satisfy (8.1), in particular, the equality
   $$F(x)=\lim_{\substack{y\downarrow x\\ y\in D}}F_D(y)\;,$$
where $F_D$ denotes the restriction of $F$ to a dense set $D$.  The denseness of $D$ guarantees that the second equality in (8.1) holds.
