I am struggling to understand the construction presented in my lecture notes of the quadratic variation $A_t$ of a continuos, square integrable $(\mathcal G_t)$-martingale $M_t$. The idea of the proof is to construct $A_\cdot$ as a suitable limit of discrete quadratic variations of $M_\cdot$, along certain random grids with mesh tending to zero.
For this purpose define a sequence of stopping times $\tau_k^n$ ($n$ controls the "mesh" of the grid) as follows:
$$\tau_k^0 = k \ \ \ \text{for $k \ge 0$}$$
and for $n \ge 1$, by induction,
$$\tau_0^n = 0$$ and for $l \ge 0$, on the event $\{ \tau_k^{n-1} \le \tau_l^n < \tau_{k+1}^{n-1}\}$, $k \ge 0$, $$\tau_{l+1}^n = \inf \left\{ t \ge \tau_l^n: |M_t - M_{\tau_l^n}| \ge \frac 1n\right\} \wedge \left(\tau_l^n + \frac 1n\right) \wedge \tau_{k+1}^{n-1}$$
Already I have some trouble with the definition; intuitively it is the first time when your process grows more than $\frac 1n$ at most at a distance of $\frac 1n$ since the previous point on the grid. I don't know why we take the second minimum though or what exactly means "on the event": what about all the $\omega$ not on the event $\{ \tau_k^{n-1} \le \tau_l^n < \tau_{k+1}^{n-1}\}$? how do you define $\tau_{l+1}^n$ for those? Also, who is $k$ exactly in the previous definition?
As a result of this confusion, I am not able to understand why the following properties are true. Lecture notes simply state that it's easy to see using continuity of $M_\cdot$ that for every $\omega \in \Omega$ we have
$$ \tau_k^n(\omega) \le \tau^n_{k+1}(\omega), \ \ \ \text{for $n,k \ge 0$}$$
This makes sense for $\omega$ in that event, but what about other $\omega$?
$$\{\tau_0^n(\omega), \tau_1^n(\omega), \dots \} \subseteq \{\tau_0^{n+1}(\omega), \tau_1^{n+1}(\omega) \dots \}$$
No idea why this would be the case.
$$\tau_k^n(\omega) \to \infty \ \ \text{as $k \to \infty$}$$
Not sure about this either.
The problem is that I cannot "visualize" the grid, I have no clue about the "spirit" of the proof. It seems to me a huge mess of symbols put there for no particularly good reason! Any help is much appreciated, even just a comment or something to make me gain intuition about this
Thanks in advance!