= Solution
Write $Z_t=\langle\kappa_t\rangle$, where the angle brackets average only over walk paths, and use $\mathbb E$ only for the environment. Every nearest-neighbor path of length $t$ has probability $(2d)^{-t}$, so the finite path sum is exactly the walk expectation. It depends only on environment layers through time $t$, and hence is $\mathcal F_t$-measurable. Its factors lie in $[1-\varepsilon,1+\varepsilon]$, so $Z_t$ is positive and integrable at each fixed time.
For any fixed path through time $t+1$, the environment variable $h(t+1,\xi_{t+1})$ is <independent> of $\mathcal F_t$ and has <mean> zero. <Conditional expectation> therefore removes its last factor. Summing over the $2d$ possible last steps of each length-$t$ path cancels the extra factor $2d$ in the path probability. Thus
$$
\boxed{\mathbb E[Z_{t+1}\mid\mathcal F_t]=Z_t,\qquad Z_0=1,\qquad \mathbb EZ_t=1.}
$$
This is the <positive random-environment path-weight martingale>. It is nonnegative and bounded in $L^1$, so the <martingale convergence theorem> gives a finite limit $\zeta\ge0$ almost surely. At this stage one can conclude $\mathbb E\zeta\le1$ by <Fatou's lemma>, but not yet equality; the second-moment bound in part (b) will prevent loss of <mean>.
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