I am reading Protter's Stochastic Integration and Differential Equations and I have difficulties with the proof of Theorem 8 of Chapter 2.3 on page 55.
The theorem states: Each $L^2$ martingale with càdlàg paths is a semimartingale.
Protter defines a semimartingale as a process $X$ such that $\forall \, t \in [0, \infty)$ the stopped process $X^t$ is càdlàg, adapted and the operator $I_{X^t}$ is continuous.
The linear operator $I_X$ is defined as $$I_X\,:S_u \to L^0 \,: H \mapsto H_0X_0 + \sum_{i=1}^n H_i (X_{T_{i+1}} - X_{T_i}),$$ where $H$ is a simple predictable process and has thus the representation $$H_t = H_0 1_{\{0\}} + \sum_{i=1}^n H_i 1_{(T_i, T_{i+1}]}, $$ with $0 = T_1 \leq \dots \leq T_{n+1} < \infty$ being a finite sequence of stopping times. The space $S_u$ of simple predictable processes is endowed with the topology of uniform convergence in $(t,\omega)$ and the space $L^0$ is topologized by convergence in probability.
So much for the setup. The only difficult part in proving the theorem is of course to show the continuity of $I_X$ when $X$ is an $L^2$ martingale.
Proof of the book: Let $X$ be an $L^2$ martingale with $X_0 = 0$ and let $H \in S$. Using Doob's Optional Sampling and the $L^2$ orthogonality of the increments of $L^2$-martingales, Protter arrives at the following inequality $$ E\{(I_X(H))^2\} \leq (\sup_{(t,\omega)}|H(t,\omega)|)^2 E\{X_\infty^2\}.$$
Question: I do understand how he arrives at the inequality, but I do not see how this inequality helps us in proving the continuity of $I_X$.
Many thanks in advance.