 Solution

A discrete-time stopping time for a filtration $({\mathcal{F}}_n)$ is a random variable $\tau$ taking values in $\{0,1,\dots \}\cup \{\infty \}$ such that $\{\tau \le n\}\in {\mathcal{F}}_n$ 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,Each event in this finite union belongs to ${\mathcal{F}}_j\subseteq {\mathcal{F}}_n$, by the definition of the natural filtration. Therefore $\tau$ is a stopping time. This verification does not require first knowing whether $\tau$ is finite almost surely.