I need some help proving the details of the proof of Lemma 6 from the following link also attached below: https://almostsuremath.com/2009/11/15/stopping-times-and-the-debut-theorem/
These facts seem straightforward however, I am having difficulty actually writing them down. I would greatly appreciate some help here.
So assuming that $X$ is a cadlag adapted process, we would like to approximate the graph $$\{(t,\omega) \in \mathbb{R}_+ \times \Omega: \Delta X_t(\omega) \neq 0\}$$ by the union of the graphs $[\tau_{s,\epsilon}]$, where we define for positive rationals $s,\epsilon$ $$\tau_{s,\epsilon} = \inf \{t \ge s: |\Delta X_t| > \epsilon\}.$$
My first question is, why is the union of the graphs $[\tau_{s,\epsilon}]$ over positive rationals $s,\epsilon$ equal to the set of time at which $\Delta X \neq 0$? The case for rational $\epsilon$'s is clear, but I am not sure how considering only the infimum over the times $\ge$ rational points is enough here. This seems obvious but I can't write down the argument for it.
Finally, when we define $$U_n = \sup \{|X_v - X_u|: u,v \in T, |v-u|<1/n\}$$ for a countable dense subset $T$ of $[s,t]$ including $t$, how does the cadlag property of $X$ guarantee that we have $$\sup_{u \in (s,t]}|\Delta X_u| = \lim_{n\to \infty} U_n?$$ Again I am not sure how the cadlag property passes over to the jumps of $X$ to give this equivalence. Also, it feels like we need the right continuity of the filtration for this to work but the post explicitly states only the completion assumption on the filtration.
These are some arguments that seem trivial when reading but has been very difficult for me to write down. I would greatly appreciate if anyone could clarify these points to me.
