This is Theorem 21 of Chapter 2 from Protter's Stochastic Integration and Differential Equations.
Here, $\mathbf{S}$ denotes the space of simple predictable processes such that $H \in \mathbf{S}$ if $H$ has a representation $$H_t = H_0 1_0(t) + \sum_{i=1}^n H_{T_i} 1_{(T_i, T_{i+1}]}$$ where $0=T_1 \le T_2 \le \cdots \le T_{n+1}<\infty$ is a finite sequence of stopping times and $H_i \in \mathscr{F}_{T_i}$ with $|H_i|<\infty $, a.s. $0 \le i \le n$.
$\mathbb{D}$ is the space of adapted, cadlag processes and $\mathbb{L}$ is the space of adapted, caglad processes.
ucp here means converge uniformly on compacts in probability. The continuous linear operator $J_X$ is defined as the stochastic integral from $\mathbb{L}_{ucp} \to \mathbb{D}_{ucp}$, where we use the extension theorem for the $\mathbb{L}_{ucp}$ space by $\mathbf{S}$.
Question: In the final sentence of the first paragraph of the proof, how can we show convergence of $(Y_{+}^k)^{\sigma_n} - Y^{\sigma_n}$ as $k \to \infty$ uniformly in $n$? We could write $Y^k = Y_0^k 1_0 + \sum_{j=1}^m Y_{S_j}^k 1_{(S_i, S_{i+1}]}$. Then we would have $Y_{+}^k = \sum_{j=1}^{m+1} Y_{S_{i-1}}1_{[S_{i-1},S_i)}$.But I cannot see how to write down $(Y_+^k)^{\sigma_n}$ from the definition of the $\sigma_n$ sample process given below since this requires intervals of $(T_i,T_{i+1}]$. It seems like we must use the property that $\Vert \sigma_n \Vert \to 0$ a.s. Also, how does $X$ being right continuous give us that $\int (Y^k - (Y_+^k)^{\sigma_n})dX_s$ tend to $0$ as $n\to \infty$?
These facts do not seem so obvious as written down in the proof but I cannot figure out how to show them. I would greatly appreciate some help.
