How I came to this:
Let $\{X_n\}_{n\in\mathbb{N}}$ be a sequence of non negative random variables, $\mathcal{F}_n=\sigma(X_l,l\leq n)$ the sequence of corresponding sigma-algebras and define $Z=\{X_n=0 \text{ for some } n\geq1\}$.
Assume there is a sequence of numbers $\varepsilon_k>0$ such that $P(Z|\mathcal{F}_n)\geq\varepsilon_k$ almost surely on $\{X_n\leq k\}$ for all $k,n\in\mathbb{N}$.
Show that almost surely either $X_n=0$ for some $n$ or $\lim_{n\to\infty}$ $X_n=\infty$.
My Problem:
I don not really understand how I can get information about $X_n$ or $\lim_{n\to\infty}$ $X_n$ and how one proves "almost surely either ... or" in a clever way.
Weak ideas so far:
One can consider $M_n=P(Z|\mathcal{F}_n)=E(\mathbb{1}_Z|\mathcal{F}_n)$ as a martingale (which seems reasonable since we get lots of helpful theory). Since $\mathbb{1}_Z$ is in $L_1$ we can show that $M_n$ is uniformly integrable. Therefore $M_n\rightarrow M_\infty$ a.s. and in $L^1$ with $L_1\ni M_\infty=E(\mathbb{1}_Z|\mathcal{F}_\infty)$ but I can't really see how I can go further from here.
Any suggestions are really appreciated. Thanks a lot!