Let $E \subset \mathbb{R}^n$ measurable. Prove that if there exist $p_0 \geq 1$ such that $f \in L^{p_o}(E) \cap L^{\infty}(E)$, then $f \in L^p(E)$ for all $p \geq p_0$ and $\|f\|_p \rightarrow \|f\|_{\infty}$ when $p \rightarrow \infty$.
I've already proved that $f\in L^p(E)$ with this conditions. The norm in the intersection space is $\|f\|_{p_0, \infty} = \|f\|_{p_0} + \|f\|_{\infty}$. Now if i set $L = \|f\|_{\infty}$ then $$\|f\|_p \leq \left( \int_E L^p \right)^{1/p}=L \cdot m(E)^{1/p} $$
is this correct? If it is, i have problems to finish the idea because i can't say that $m(E) \rightarrow 1$ because maybe $m(E)=+\infty$
If $\epsilon>0$ is smaller than $L=\|f\|_\infty$ then the set $A_\epsilon:=\{x\in E: |f(x)|>L-\epsilon\}$ satisfies $0<m(A_\epsilon)<\infty$ (why?) and $$ \int_E |f|^p\,dm\ge\int_{A_\epsilon}|f|^p\,dm\ge (L-\epsilon)^pm(A_\epsilon). $$ Consequently $$ \|f\|_p\ge(L-\epsilon)[m(A_\epsilon)]^{1/p}. $$ The rest of the way is clear sailing.