Since $|Z| \leq C$ almost surely, result R4127: Real exponentiation function with unsigned exponent is isotone on unsigned reals implies that $|Z|^p \leq C^p$ almost surely. Applying result R1818: Isotonicity of real expectation, we thus get \begin{equation} \mathbb{E} |Z|^p \leq \mathbb{E} C^p = C^p < \infty \end{equation}