I have recently been trying to prove the following using only knowledge from chapters 1 through 6 of Rudin's "Principles of Mathematical Analysis". In other words, I cannot use the concept of measure.
$\lim_{p\rightarrow \infty}(\int_ {0}^{1}|f(x)|^p)^{1/p}=\max_{x\in[0,1]}|f(x)|$ if $f:[0,1]\rightarrow \mathbb{R}$ is continuous on its domain
where $| \cdot|$ is the absolute value function. However, I am unsure where to start and most proofs I have found online rely on concepts not covered in the mentioned chapters.
My attempt involved writing $\lim_{p\rightarrow \infty}(\int_ {0}^{1}|f(x)|^p)^{1/p}$ in the form $\lim_{p\rightarrow \infty}(\lim_{n \rightarrow \infty}(\frac{1}{n}\sum_{i=1}^{n}|f(\frac{i}{n})|^p))$. From there, I tried to use properties of Darboux lower and upper sums and integrals to reach an $\epsilon-\delta $ limit conclusion but I was not able to make much progress.
Could anyone point me in the right direction? Any hints, tips, or proof outlines would be greatly appreciated.
