= Solution
{c}
A discrete-time <stopping time> for a <filtration> $(\mathcal F_n)$ is a <random variable> $\tau$ taking values in $\{0,1,\ldots\}\cup\{\infty\}$ such that $\{\tau\leq n\}\in\mathcal F_n$ for every $n$. It means that whether stopping has occurred by time $n$ can be determined from the information available then.
For the first elimination time,
$$
\{\tau\leq n\}=\bigcup_{j=0}^n\{X_jY_jZ_j=0\}.
$$
Each <event> in this finite <union> belongs to $\mathcal F_j\subseteq\mathcal F_n$, by the definition of the <natural filtration>. Therefore \b[$\tau$ is a stopping time]. This verification does not require first knowing whether $\tau$ is finite <almost surely>.
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