Bounddeness in $L^p$ norm implies boundedness in infinity norm

Question

Assume $f\in L^p$ and $||f||_p\le C$ for every $1<p<\infty$. Prove that $f\in L^\infty$ and $||f||_\infty\le C$.

I know that from a problem in Folland (Problem 7 on Page 187), if $f\in L^\infty$, then the infinity norm is the limit of the p norm, but I don't know how to show $f\in L^\infty$.

Answer 1

$fI_{\{x:|f(x)| \leq n\}}$ is an $L^{\infty}$ function so by the problem you have quoted we get $\|fI_{\{x:|f(x)| \leq n\}}\|_{\infty} \leq C$. If you let $n \to \infty$ you see that $f \in L^{\infty}$ and $\|f\|_{\infty} \leq C$.