Let $(X,\mathfrak{X},\mu)$ be a finite measure space and let $f \in L^p(\mu)$. Use Holder's inequality to show that: \begin{equation} f \in L^r(\mu) \end{equation}
for $1 \leq r \leq p < \infty$.
Applying Holder's inequality show that: \begin{equation} ||f||_r \leq ||f||_p \mu(X)^s \end{equation}
where $s = (1/r) - (1/p)$. Thus if $\mu(X) = 1$ then $||f||_r \leq ||f||_p$.
A finite measure space satisfies $\mu(X) < \infty$.
Furthermore, Holder's inequality states that: \begin{equation*} \int |fg| \,d\mu \leq \left( \int |f|^p \,d\mu \right)^{1/p} \left(\int |g|^q \,d\mu \right)^{1/q} \end{equation*}
with $f,g$ measurable and $ 1 < p < \infty$ and $q = \frac{p}{p-1}$
Since $f \in L^p(\mu)$ we know that:
\begin{equation} \left( \int |f|^p \,d\mu \right)^{1/p} < \infty \end{equation}
However, I do not see how to apply it in this case, as there are restrictions on $q$, which do not appear in the result I am trying to prove. Any help is much appreciated. I am stuck.