Common refinement for simple predictable processes

Question

I'm trying to show that for a semimartingale $X$, the stochastic integral $\mathcal{I}_X : \mathcal{S}_{\text{ucp}} \rightarrow \mathbb{D}_{\text{ucp}}$ is cauchy continuous, since the idea is to extend its domain to $\mathbb{L}$. I know that it is already continuous, but Protter's book says that we can extend continuous linear functions defined on dense sets. However, I think that the function must preserve cauchy sequences in order to do so. So I am trying to use the definition of the integral on simple predictable processes and the ucp topology directly to show that the integral does conserve cauchy sequences. I think this comes down to comparing simple processes. Is there a way to get a common refinement like in Riemann integration, except now with stopping times so we can write the difference in a canonical representation? For example if $H_1 = Z_0\mathbf{1}_{\{0\}} + \sum\limits_{k=1}^{n-1} Z_k\mathbf{1}_{(T_k, T_{k+1}]}$ and $H_2 = Y_0\mathbf{1}_{\{0\}} + \sum\limits_{k=1}^{m-1} Y_k\mathbf{1}_{(S_k, S_{k+1}]}$, is there some notion of a common refinement of the stopping times $S_k, T_k$ ... maybe $L_k$ such that $H_1-H_2 = A_0\mathbf{1}_{\{0\}} + \sum\limits_{k=1}^{q-1} A_kL_{(L_k, L_{k+1}]}$