I'm studying Stochastic calculus and applications, Cohen. He claims that
As $M \in \mathcal{H}^{2,d}$, we know $\lim_k \sum_{j=1}^k \mathbb{E}[(\Delta M_{T_n})^2] < \infty$.
Note that $T_n$ is all jump times of $M$, which is either totally inaccessible or predictable time. But how this be true? I think it is true for just $\mathcal{H}^2$ which is the space of square integrable martingale meaning $$\mathbb{E}[\sup_{t \in [0,\infty)} |M_t|^2] < \infty$$ It may not have integrable variation. This content is in the proof of Theorem 10.2.14 of Cohen's book. Any help?