A question related to $L^p$ space

Question

Let $E⊂\mathbb{R}^n$ measurable. Prove that if there exist $p_{0}≥1$ such that $f∈L^{p_{0}}(E)∩L^∞(E)$, then $f∈L^p(E)$ for all $p≥p_0$.

How do I prove this claim? Any help will be appreciated.

Answer 1

If $p > p_0$ and $f \in L^{p_0} \cap L^\infty$ then $|f|^p = |f|^{p - p_0}|f|^{p_0} \le \|f\|_\infty^{p - p_0}|f|^{p_0}$ almost everywhere.