What techniques for proving that a stopping time is finite almost surely? In Shreve Stochastic Calculus For Finance II, the proof for $P [ \tau_m < \infty ] = 1 $, where $\tau$ is defined as $\min \{ t \geq 0 ; W_t = m \}$ whith $W_t$ a standard Brownian Motion, envolves the properties of the exponential martingale $Z_t = e^{\sigma W_t - \frac{1}{2}\sigma^2t}$.
I was wondering (generally speaking), what are the other "techniques" used to show that a stopping time is finite almost surely?. Is there any "classic"/well know exercises that illustrate some techniques in other configurations?
 A: One useful lemma is William's $``$ Sooner than later lemma$"$(Steele's Stochastic Calculus). 

Lemma: Suppose for the stopping time $\tau$. There exists $N$ and $\epsilon>0$ such that for all $n\geq 0$,  $\mathbb{P}(\tau \leq n+N|\mathscr{F}_n )\geq\epsilon$. Then $\mathbb{P}(\tau <\infty)=1$ and $E(\tau^{p})<
\infty$ for all $p\geq 1$. 

1. This can be used to show that $\tau<\infty$ almost surely, when $X_n$ is a random walk (simple or biased) and $\tau= \inf \{k| X_k\not \in (a,b)\}$. Since when $X_n\in(a,b)$ whenever $X_n$ takes $b-a$ to the right $X_n$ exists $(a,b)$.
2. The same technique can be if $X_n$ is replaced by $B_t$ the Brownian motion. In this case the lemma can be applied using the reflection principle. 
Remark: this lemma cannot be applied to your stopping time as it does not  have bounded moments
Proof of Lemma: We will show by induction that $\mathbb{P}(\tau >kN)<(1-\epsilon)^k$. The base case $k=1$, is immediate by using the given inequality for $n=0$. We do that the inductive step.
\begin{align}
\mathbb{P}(\tau >(k+1)N)&=\mathbb{P}(\tau >(k+1)N,\tau >kN) \\
&=\mathbb{P}(\tau >(k+1)N|\tau >kN)\mathbb{P}(\tau >kN)\\
&=\mathbb{P}(\tau >kN+N|\tau >kN)\mathbb{P}(\tau >kN)\\
&<(1-\epsilon)(1-\epsilon)^k\\
&=(1-\epsilon)^{k+1}
\end{align}
The $\mathbb{P}(\tau<\infty)=1$ is immediate from Borel-Cantelli, and for the bounded moment part we can do the following 
\begin{align}
E(\tau^{p})&=\int_{t\geq 0}pt^{p-1}\mathbb{P}(\tau\geq t) dt\\
&=\sum_{k\geq 0}\int_{(k+1)N\geq u>kN}pu^{p-1}\mathbb{P}(\tau\geq u)du\\
&\leq \sum_{k\geq 0}\int_{(k+1)N\geq u>kN}p((k+1)N)^{p-1}\mathbb{P}(\tau> kN)du\\
&=\sum_{k\geq 0}pN^{p}(k+1)^{p-1}\mathbb{P}(\tau> kN)\\
&<\sum_{k\geq 0}pN^{p}(k+1)^{p-1}(1-\epsilon)^k<\infty
\end{align}
