 Solution (source code)

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= Solution
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A discrete <stopping time> is a random variable $T:\Omega\to\mathbb N_0\cup\{\infty\}$ such that $\{T\leq n\}\in\mathcal F_n$ for every $n$. The convention is $\inf\varnothing=\infty$.

For the <hitting time> in question,
$$
\{T_k\leq n\}=\bigcup_{j=0}^n\{S_j=k\}.
$$
Each event in the union is in $\mathcal F_j\subseteq\mathcal F_n$ because $S_j$ is adapted. This proves \b[$T_k$ is a stopping time] for every positive integer $k$.

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