In Jean-François Le Gall's Brownian Motion, Martingales, and Stochastic Calculus, the author defines the stochastic integral for a continuous local martingale $M$ in Chapter 5, which is defined as $H \cdot M$ a continuous local martingale with initial value $0$, satisfying $$ \langle H \cdot M, N \rangle = H \cdot \langle M, N\rangle $$
for any continuous local martingale. (here $H \in L_{loc}^2 (M)$)
I am a bit confused about the construction of $H \cdot M$.
The author first defines a sequence of stopping times $(T_n)_{n\geq 1}$ that reduce $M$ and defines
$$ (H \cdot M)^{T_n} = H \cdot M^{T_n} $$
which makes sense by the construction of stochastic integral in continuous martingale case (included below as Theorem 5.4) and he also claims since $(H \cdot M)_t = \lim (H\cdot M^{T_n})_t$, the constructed $H\cdot M$ is continuous and adapted.
Questions:
I fail to see why the limit is well-defined, while the author gives a justification according to the stopping time property in Theorem 5.4, we have $H\cdot M^{T_n}=(H\cdot M^{T_m})^{T_n}, \forall m >n$.
Also, I have read the relevant section in Continuous Martingales and Brownian Motion by Yor & Revuz, it seems that the authors claim that specifying $H\cdot M$'s values on $[0, T_n]$ is sufficient, I wonder why that is true (I understand the stopping time property ensures the definitions are consistent on intervals $[0, T_n]$)
Notations:
$L^2(M)$: The set of progressive processes $H$ such that $E\left[\int_0^\infty H_s^2 d\langle M, M \rangle_s\right]<\infty$
$L^2_{loc}(M)$: The set of progressive processes $H$ such that $\int_0^t H_s^2 d\langle M, M \rangle_s<\infty, \forall t \geq 0$ almost surely
$\mathbb{H}^2$: the space of all continuous martingales bounded in $L^2$ ($\sup_{t\geq 0} E[X_t^2] < \infty$)
$T_n$: in the context above, $T_n$ is defined as $\inf\{t\geq 0: \int_0^t (1+H_s^2)d \langle M, M \rangle_s\geq n\}$
Theorem 5.4 Let $M \in \mathbb{H}^2$. For every elementary process $H$ of the form $$ H_s(\omega) = \sum_{i=0}^{p-1} H_{(i)} (\omega) \mathbf{1}_{(t_i, t_{i+1}]}(s) $$
the formula
$$ (H\cdot M)_t = \sum_{i=0}^{p-1} H_{(i)}(M_{t_{i+1} \land t} - M_{t_i \land t}) $$
defines a process $H\cdot M\in \mathbb{H}^2$. The mapping $H\mapsto H\cdot M$ extends to an isometry from $L^2(M)$ to $\mathbb{H}^2$. Furthermore, $H\cdot M$ is the unique martingale in $\mathbb{H}^2$ that satisfies $\forall H \in L^2 (M)$
$$ \langle H \cdot M, N \rangle = H \cdot \langle M, N\rangle $$
In addition, if $T$ is a stopping time, we have $$ (\mathbf{1}_{[0,T]}H) \cdot M = (H\cdot M)^T = H \cdot M^T $$