Solution
= Solution
{c}
Use $s$ for the dummy integration variable in the printed integral. The <Itô isometry> gives
$$
\mathbb E|z_t^{(n+1)}|^2=C\int_0^t\mathbb E|z_s^{(n)}|^2\,ds
\leq\frac{C^{n+1}}{n!}\int_0^t s^n\,ds
=\boxed{\frac{C^{n+1}t^{n+1}}{(n+1)!}.}
$$
The assumed second-moment bound also proves the integrability needed to apply the <Itô isometry>. The extra time integration is the source of the factorial improvement.