Set \begin{equation} r := \frac{p}{q} \quad \text{ and } \quad s := \frac{p}{p - q} \end{equation} Since $p > q$, then $p - q > 0$ and \begin{equation} \frac{1}{r} + \frac{1}{s} = \frac{q + p - q}{p} = \frac{p}{p} = 1 \end{equation} Consider now the real functions $F, G : X \to \mathbb{R}$ given by \begin{equation} F(x) = |f(x)|^q \quad \text{ and } \quad G(x) = 1 \end{equation} Applying R82: Hölder's inequality with the conjugate exponents $r$ and $s$, one has \begin{equation} \begin{split} \int_X |f|^q \, d \mu = \Vert F G \Vert_{L^1} & \leq \Vert F \Vert_{L^r} \Vert G \Vert_{L^s} \\ & = \left( \int_X |f|^{p} \, d \mu \right)^{\frac{q}{p}} \left( \int_X \, d \mu \right)^{\frac{p - q}{p}} = \Vert F \Vert_{L^p}^q \mu(X)^{\frac{p - q}{p}} < \infty \end{split} \end{equation} Exponentiating both sides to power $1 / q$, we can then conclude \begin{equation} \left( \int_X |f|^q \, d \mu \right)^{1 / q} < \infty \end{equation} $\square$