I am very puzzled with the following theorem and proof from Protter's Stochastic Integration and Differential Equations.
Setting: Below, $\mathbf{S}$ consists of processes $H$ such that $H = H_0 1_0 + \sum_{i=1}^n H_i 1_{(T_i, T_{i+1}]}$ where $0=T_1 \le T_2 \le \cdots \le T_{n+1} < \infty$ is a finite sequence of stopping times, $H_i \in \mathscr{F}_{T_i}$ with $|H_i| < \infty$ a.s.
$\mathbb{L}$ denotes the space of caglad adapted processes and $\mathbb{D}$ the space of cadlag adapted processes.
And we consider the ucp topology, i.e. a sequence of processes $H^n$ converges to a process $H$ uniformly on compacts in probability (ucp) if for each $t \ge 0$, $\sup_{0 \le s \le t} |H_s^n - H_s|$ converges to $0$ in probability.
We give the space of caglad and cadlag this ucp topology, then we are able to define given a caglad adapted process $H$, and a semimartingale $X$, $H \cdot X$, the stochastic integral of $H$ with respect to $X$, as the ucp limit of $H^k \cdot X$, for processes $H^k \in \mathbf{S}$ which converges to $H$ in ucp.
Main Question: The proof first asserts that $\Delta (H\cdot X)_s = H_s (\Delta X_s)$ up to indistinguishability.
I think we get this because: For $H \in \mathbf{S}$, the result is clear. So for $H \in \mathbb{L}$, we can find a $H^n$ which converges to $H$ in ucp.
I can see that this gives for each $t \ge 0,$ a further subsequence, say $H^k$ which converges to $H$ uniformly on $[0,t]$ a.s.
Hence this ensures that $H^k \Delta X_t$ converges a.s. to $H \Delta X_t$.
Meanwhile, we also have that $H^k \Delta X_t = \Delta (H^k \cdot X)_t$. We have, a.s., $|(H\cdot X)_s - (H^k \cdot X)_s| \to 0$ uniformly on $[0,t]$. Hence, we also get $(H^k \cdot X)_{s-} \to (H \cdot X)_{s-}$ a.s. for each $s$, and hence we get a.s. convergence of $\Delta (H^k \cdot X)_s $ to $\Delta(H \cdot X)_s$. Now we use the uniqueness of limits to conclude that $(H \Delta X)_t = \Delta(H\cdot X)_t$.
But how do we get equivalence up to indistinguishability from path regularity? I know this is the case if two processes are right continuous or left continuous, but we cannot assume such rproperty for jump processes.
I have been struggling with the details here, and I would greatly appreciate any help.
