Since $\mathbb{E} |Z|^p$ is finite, then so is its exponentiation $(\mathbb{E} |Z|^p)^{1 / p}$. Since $p > q$, result R3571: Random Lebesgue P-norm inherits finiteness from higher exponents now states that $(\mathbb{E} |Z|^q)^{1 / q}$ is also finite. Exponentiating this finite quantity by $q$, we find that $\mathbb{E} |Z|^q < \infty$. $\square$