I'm reading Stochast integration and stochastic differential equation by Protter. In particular I have a question about Theorem 10 in chapter 2.4. Here Protter defines a simple predictable processes $H$ if it has a representation $$H_t=H_0\mathbf1_{\{0\}}+\sum_{i=1}^nH_i\mathbf1_{(T_i,T_{i+1}]}(t)$$ where $0=T_1\le \dots\le T_n<\infty$ are stopping times and $H_i\in\mathcal{F}_{T_i}$ with $|H_i|<\infty$ a.s. The collection of all such processes is denoted with S. In Theorem 10 he wants to prove that S is dense in the space of all adapted caglad processes, denoted with $\mathbb{L}$, w.r.t. ucp convergence.
The goal is therefore to approximate $Y\in\mathbb{L}$ with a sequence in S. First one can reduce the claim to $Y$ which are additionally bounded. He then looks at the right continuous "version" of $Y$, i.e. $$Z:=\lim_{u\downarrow t}Y_u$$
Clearly this is an adapted and cadlag process. Now the key point of the proof is to define the following sequence of stopping times for $\epsilon>0$: $$T^\epsilon_0:=0$$ $$T^\epsilon_{n+1}:=\inf\{t:t>T^\epsilon_n \text{ and }|Z_t-Z_{T^\epsilon_n}|>\epsilon\}$$
Moreover $$Z^\epsilon:=\sum_n Z_{T^\epsilon_n}\mathbf1_{[T^\epsilon_n,T^\epsilon_{n+1})}$$ Protter says $Z^\epsilon$ converges uniformly to $Z$ as $\epsilon\to 0$. In what sense does he mean uniformly and how is it established?
Assuming this he defines the left continuous process $$ U^\epsilon=Y_0\mathbf1_{\{0\}}\sum_n Z_{T^\epsilon_n}\mathbf1_{(T^\epsilon_n,T^\epsilon_{n+1}]}$$ How does the proceeding implies $U^\epsilon\to Y_0\mathbf1_{\{0\}}+Z_{-}=Y$ in the ucp sense?