The question is: Let $ (X,\mathcal M,\mu) $ be an arbitrary measure space. Let $f$ be a function in $L^r$ for some $0<r<\infty$. Show that $||f||_p$ converges to $||f||_\infty$ as $p\to \infty$.
Now I have shown that this is true under the additional assumptions that $||f||_\infty<\infty$, and that $f$ is a simple function. Now let $f$ be nonnegative and let $s_n$ be increasing to $f$ where $s_n$'s are simple functions with finite measure support. But how could I pass to limits? What I am arguing is: \begin{align} \lim_{p\to\infty}||f||_p&=\lim_{p\to\infty}\lim_{n\to\infty}||s_n||_p, \text{ (Monotone Convergence Theorem)}\\ &=\lim_{n\to\infty}\lim_{p\to\infty}||s_n||_p\\ &=\lim_{n\to\infty}||s_n||_\infty=||f||_\infty \end{align} But I cannot justify switching limits in the second line. How could I do this? Or alternatively, is there another proof that avoids the progressive argument?
