Hello!! In a space with measure $1$, $||f||_p$ is a oncreasding function with respect of $p$. To show that $\lim_{p \rightarrow \infty} ||f||_p=||f||_{\infty}$ we have to show that $||f||_{\infty}$ is the supremum, right??
To show that, we assume that $||f||_{\infty}-\epsilon$ is the supremum.
From the esential supremum we have that $m(\{|f|>||f||_{\infty}-\epsilon\})=0$.
So, we have to show that $m(\{|f|>||f||_{\infty}-\epsilon\})>0$.
Let $A=\{|f|>||f||_{\infty}-\epsilon\}$.
We have that $\int_A |f|^p \leq \int |f|^p \leq ||f||_{\infty}^p$.
