= Solution
Define the <stopped martingale>
$$
\boxed{M_n=1-S_{n\wedge T_1}.}
$$
It is nonnegative: before the first visit to $1$, the nearest-neighbour <random walk> is at most $0$, and afterwards the stopped position is $1$. Each $M_n$ is integrable, with $|M_n|\leq n+1$. Moreover
$$
M_{n+1}-M_n=-\mathbf1_{\{T_1>n\}}X_{n+1}.
$$
The indicator is $\mathcal F_n$-measurable and the increment has conditional mean zero, so $M_n$ is a <martingale> with $\mathbb E M_n=M_0=1$.
Part (a) shows $T_1<\infty$ almost surely, so every such path eventually has $M_n=0$. Thus $M_n\to0$ almost surely. But $\mathbb E|M_n-0|=1$ for every $n$, so there is no <convergence in L1>. Any possible <L1 convergence> limit would agree with the almost sure limit. This is a \b[nonnegative <martingale> with <almost sure convergence> but no <convergence in L1>], and hence it is not <uniformly integrable>.
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