Past exam of the martingales and financial mathematics course of the University of Oxford / 2009 / A / 3 / b / ii

A discrete stopping time is a random variable $T:\Omega \to {\mathbb{N}}_0\cup \{\infty \}$ such that $\{T\le n\}\in {\mathcal{F}}_n$ for every $n$. The convention is $\inf \varnothing =\infty$.

For the hitting time in question,Each event in the union is in ${\mathcal{F}}_j\subseteq {\mathcal{F}}_n$ because $S_j$ is adapted. This proves $T_k$ is a stopping time for every positive integer $k$.