$L^p$ and $L^\infty$ So I am trying to prove that for a set $E$ of finite measure, and for $1 \leq p < \infty$, $||f||_p \leq (m(E))^{1 - 1/p}||f||_{\infty}$.  But I think I have proved the wrong thing.  Can you help me see where I went wrong?
My proof is something like

$$||f||_p =\left(\int_E |f|^p\right)^{1/p} \leq \left(\int_E ||f||_{\infty}^p\right)^{1/p} = \left(||f||_{\infty}^p \int_E 1\right)^{1/p} = ||f||_{\infty} (m(E))^{1/p},$$

which is not what was asked for in the problem.
Thanks!

What you get is true but not the wanted inequality. But you can write, assuming that $f\in L^{\infty}$ $|f|^p=|f|^{p-1}|f|\leq ||f||_{\infty}^{p-1}|f|$ then apply Hölder's inequality.
