Why is it true that $\lim_{n\to\infty} \left(\int_a^b \left|f(x)\right|^ndx\right)^{1/n} = \text{sup}\{\left|f(x)\right| : x \in [a,b]\}$?
If $f(x)$ is continuous on $[a,b]$ and $M=\max \; |f(x)|$, is $M=\lim \limits_{n\to\infty} \left(\int_a^b|f(x)|^n\,\mathrm dx\right)^{1/n}$?