= Solution
{c}
Boundedness makes each $N_u$ Q-integrable. For $s\leq u\leq t$, the density formula and conditioning first on $\mathcal F_u$ give
$$
\mathbb E_Q[N_u\mid\mathcal F_s]
=\frac{\mathbb E_P[N_uZ_t\mid\mathcal F_s]}{Z_s}
=\frac{\mathbb E_P[N_uZ_u\mid\mathcal F_s]}{Z_s}
=\frac{N_sZ_s}{Z_s}=N_s.
$$
The middle equality uses that $N_u$ is $\mathcal F_u$-measurable and $Z$ is a martingale; the next uses the assumed martingale property of $NZ$. Thus \b[$N$ is a Q-martingale].
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