I am self-studying the following material, whose source is https://sayanmuk.github.io/StochasticAnalysisManifolds.pdf
However, I am stuck with one step of the proof of the only theorem that follows after the definitions:
Definition 1.1.6. Let $(\Omega, \mathcal{F}^*, P)$ be a filtered probability space and $\tau$ an $\mathcal{F}^*$-stopping time. A continuous process $X$ defined on the stochastic time interval $[0, \tau)$ is called an $\mathcal{F}^*$-semimartingale up to $\tau$ if there exists a sequence of $\mathcal{F}^*$-stopping times $\tau_n \uparrow \tau$ such that for each $n$ the stopped process $X_{\tau_n} = \{X_t \wedge \tau_n, t \geq 0\}$ is a semimartingale in the usual sense.
Definition 1.1.7. A semimartingale $X$ up to a stopping time $\tau$ is a solution of SDE($\sigma$, $Z$, $X_0$) if there is a sequence of stopping times $\tau_n \uparrow \tau$ such that for each $n$ the stopped process $X_{\tau_n}$ is a semimartingale and $$X_t \wedge \tau_n = X_0 + \int_0^{t \wedge \tau_n} \sigma(X_s) \, dZ_s, \quad t \geq 0.$$
We are now in a position to show that there is a unique solution $X$ to SDE($\sigma$, $Z$, $X_0$) up to its explosion time $e(X)$.
Theorem 1.1.8. Suppose that we are given (i) a locally Lipschitz coefficient matrix $\sigma : \mathbb{R}^N \rightarrow \mathbb{M}(N, l)$; (ii) an $\mathbb{R}^l$-valued $\mathcal{F}^*$-semimartingale $Z = \{Z_t, t \geq 0\}$ on a filtered probability space $(\Omega, \mathcal{F}^*, P)$; (iii) an $\mathbb{R}^N$-valued, $\mathcal{F}_0$-measurable random variable $X_0$. Then there is a unique $\mathbb{W}(\mathbb{R}^N)$-valued random variable $X$ which is a solution of SDE($\sigma$, $Z$, $X_0$) up to its explosion time $e(X)$.
Proof. We first assume that $X_0$ is uniformly bounded: $|X_0| \leq K$. For a fixed positive integer $n \geq K$ let $\sigma_n : \mathbb{R}^d \rightarrow \mathbb{M}(N, l)$ be a globally Lipschitz function such that $\sigma_n(z) = \sigma(z)$ for $|z| \leq n$. By Theorem 1.1.3, SDE($\sigma_n$, $Z$, $X_0$) has a solution $X_n$. We have $X_n^t = X_{n+1}^t$ when $t \leq \tau_n$ where $\tau_n = \inf \{t \geq 0 : |X_n^t| \text{ or } |X_{n+1}^t| = n\}$.
This follows from the uniqueness part of Theorem 1.1.3 because $\sigma_n(z) = \sigma_{n+1}(z)$ for $|z| \leq n$ and the two semimartingales $X_{n,\tau_n}$ and $X_{n+1,\tau_n}$ ($X_n$ and $X_{n+1}$ stopped at $\tau_n$ respectively) are solutions of the same equation SDE($\sigma_n$, $Z_{\tau_n}$, $X_0$). Now $\tau_n$ is the first time the common process reaches the sphere $\partial B(n) = \{z \in \mathbb{R}^N : |z| = n\}$. In particular, we have $\tau_n \leq \tau_{n+1}$. Let $e = \lim_{n \to \infty} \tau_n$, and define a semimartingale $X$ up to time $e$ by $X_t = X_{n,t}$ for $0 \leq t < \tau_n$. Then $\tau_n$ is the first time the process $X$ reaches the sphere $\partial B(n)$. From $$X_{n,t} = X_0 + \int_0^t \sigma_n(X_s) \, dZ_s, \quad X_t \wedge \tau_n = X_{n,t},$$ and $\sigma_n(X_s) = \sigma(X_s)$ for $s \leq \tau_n$, we have $$X_t \wedge \tau_n = X_0 + \int_0^{t \wedge \tau_n} \sigma (X_s) \, dZ_s,$$ which means that $X$ is a solution of SDE($\sigma$, $Z$, $X_0$) up to time $e$.
We now show that $e$ is the explosion time of $X$, i.e., $$\lim_{t \to e^+} |X_t| = \infty \text{ on } \{e < \infty\}.$$ Equivalently, we need to show that for each fixed positive $R \geq K$, there exists a time (not necessarily a stopping time) $t_R < e$ such that $|X_t| \geq R$ for all $t \in [t_R, e)$.
The idea of the proof is as follows. Because the coefficients of the equation are bounded on the ball $B(R + 1)$, $X$ needs to spend at least a fixed amount of time (in an appropriate probabilistic sense) when it crosses from $\partial B(R)$ to $\partial B(R + 1)$. If $e < \infty$, this can happen only finitely many times. Thus after some time, $X$ either never returns to $B(R)$ or stays inside $B(R+1)$ forever; but the second possibility contradicts the facts that $|X_{\tau_n}| = n$ and $\tau_n \uparrow e$ as $n \uparrow \infty$.
To proceed rigorously, define two sequences $\{\eta_n\}$ and $\{\zeta_n\}$ of stopping times inductively by $\zeta_0 = 0$ and $$\eta_n = \inf \{t > \zeta_{n-1} : |X_t| = R\}, \quad \zeta_n = \inf \{t > \eta_n : |X_t| = R + 1\},$$ with the convention that $\inf \{\emptyset\} = e$. If $\zeta_n < e$, the difference $\zeta_n - \eta_n$ is the time $X$ takes to cross from $\partial B(R)$ to $\partial B(R + 1)$ for the $n$th time. By Itô's formula applied to the function $f(x) = |x|^2$ we have $$|X_t|^2 = |X_0|^2 + N_t + \int_0^t \Psi_s \, dQ_s, \quad t < e,$$ where $$N_t = 2 \int_0^t \sigma_i^\alpha(X_s)X_i^s \, dM_s^\alpha,$$ $$\Psi_s = 2\sigma_i^\alpha(X_s)X_i^s \, dA_s^\alpha dQ_s + \sigma_i^\alpha(X_s)\sigma_i^\beta(X_s) \, d\langle M^\alpha, M^\beta \rangle_s dQ_s.$$
The definition of $Q$ is given in (1.1.3). It is clear that $$N_t^* = \int_0^t \Phi_s \, dQ_s, \quad \Phi_s = 4\sigma_i^\alpha(X_s)\sigma_j^\beta(X_s)X_i^sX_j^s \, d\langle M^\alpha, M^\beta \rangle_s dQ_s.$$
By the well-known Lévy's criterion, there exists a one-dimensional Brownian motion $W$ such that $$N_u + \eta_n - N_{\eta_n} = W(hN_iu + \eta_n - hN_i\eta_n).$$
When $\eta_n \leq s \leq \zeta_n$ we have $|X_s| \leq R + 1$. Hence there is a constant $C$ depending on $R$ such that $|\Psi_s| \leq C$ and $|\Phi_s| \leq C$ during the said range of time. From (1.1.9) we have $$1 \leq |X_{\zeta_n}|^2 - |X_{\eta_n}|^2 = W(hN_{\zeta_n} - hN_{\eta_n}) + \int_{\zeta_n}^{\eta_n} \Psi_s \, dQ_s \leq W^* (hN_{\zeta_n} - hN_{\eta_n}) + C (Q_{\zeta_n} - Q_{\eta_n}),$$
where $W^*_t = \max_{0 \leq s \leq t} |W_s|$. We also have $hN_{\zeta_n} - hN_{\eta_n} \leq C(Q_{\zeta_n} - Q_{\eta_n})$. Now it is clear that the events $\zeta_n < e$ and $Q_{\zeta_n} - Q_{\eta_n} \leq (Cn)^{-1}$ together imply that $$W^*_{1/n} \geq 1 - \frac{1}{n} \geq \frac{1}{2}.$$
Therefore, $$P(\zeta_n < e, Q_{\zeta_n} - Q_{\eta_n} \leq (Cn)^{-1}) \leq P(W^*_{1/n} \geq 1/2) = \frac{1}{2} \leq \frac{r}{2},$$
where $r = \frac{2n}{\pi} \int_0^{1/2} e^{-nu^2/2} \, du \leq \frac{1}{4}$. By the Borel-Cantelli lemma, either $\zeta_n = e$ for some $n$ or $\zeta_n < e$ and $Q_{\zeta_n} - Q_{\eta_n} \geq (Cn)^{-1}$ for all sufficiently large $n$. The second possibility implies that $$Q_{\zeta_n} \geq \sum_{m=1}^{n} (Q_{\zeta_m} - Q_{\eta_m}) \rightarrow \infty,$$
which in turn implies $\zeta_n \uparrow \infty$. Thus if $e < \infty$, we must have $\zeta_n = e$ for some $n$. Let $\zeta_{n_0}$ be the last time strictly less than $e$. Then $X$ never returns to $B(R)$ for $\zeta_{n_0} \leq t < e$. This shows that $e$ is indeed the explosion time and $X$ is a solution of SDE($\sigma$, $Z$, $X_0$) up to its explosion time.
Question: In this part:
Therefore, $$P(\zeta_n < e, Q_{\zeta_n} - Q_{\eta_n} \leq (Cn)^{-1}) \leq P(W^*_{1/n} \geq 1/2) = \frac{1}{2} \leq \frac{r}{2},$$
where $r = \frac{2n}{\pi} \int_0^{1/2} e^{-nu^2/2} \, du \leq \frac{1}{4}$. By the Borel-Cantelli lemma, either $\zeta_n = e$ for some $n$ or $\zeta_n < e$ and $Q_{\zeta_n} - Q_{\eta_n} \geq (Cn)^{-1}$ for all sufficiently large $n$
The author proves the probability of the $\lim sup_{n\to\infty} P(\zeta_n < e, Q_{\zeta_n} - Q_{\eta_n} \leq (Cn)^{-1})=0$, however, how can the author infer that by the Borel-Cantelli lemma, either $\zeta_n = e$ for some $n$ or $\zeta_n < e$ and $Q_{\zeta_n} - Q_{\eta_n} \geq (Cn)^{-1}$ for all sufficiently large $n$? I know the contrary of $\{\zeta_n < e, Q_{\zeta_n} - Q_{\eta_n} \leq (Cn)^{-1}\}$ of should be $1$, but that would be $\{\zeta_n > e, Q_{\zeta_n} - Q_{\eta_n} \geq (Cn)^{-1}\}$ instead of the two possibilities that are presented. How can this step be explained that $\zeta_n=e$ for some big n?
Thanks in advance.
He shows that the events
$$A_{n}:=\{\zeta_n < e, Q_{\zeta_n} - Q_{\eta_n} \leq (Cn)^{-1}\}$$
are summable $\sum P(A_{n})\leq \sum r_{n}$ for $r_{n} = \frac{2n}{\pi} \int_0^{1/2} e^{-nu^2/2} \, du \leq \frac{1}{4}$. Therefore, by Borel-Cantelli for each omega we have some $N_{0}(\omega)$ so that for all $n\geq N_{0}(\omega)$ the $A_{n}^{c}$ is true. We split into disjoint sets:
$$\{\zeta_n \geq e\}\cup\{ Q_{\zeta_n} - Q_{\eta_n} > (Cn)^{-1}\}$$
$$=\{\zeta_n \geq e,Q_{\zeta_n} - Q_{\eta_n} \leq (Cn)^{-1}\}\sqcup \{\zeta_n \geq e, Q_{\zeta_n} - Q_{\eta_n} > (Cn)^{-1}\}\sqcup \{\zeta_n < e, Q_{\zeta_n} - Q_{\eta_n} > (Cn)^{-1}\}.$$
So we indeed have the claim.