Pulling limit inside $L^\infty$ norm?

Question

I want to prove that $||\lim_{n\rightarrow\infty} f_n||_\infty\leq \lim_{n\rightarrow\infty}||f_n||_\infty$ if both limits exist. I know that for $L^p$ one can use Fatou's lemma, but I'm not sure how to do it in the $L^\infty$ case.

Answer 1

$L^\infty$ is much simpler than $L^p$ in certain aspects. For a.e. $x$ and each $n$, $|f_n(x)| \le ||f_n||_\infty$. Hence, $\lim_n |f_n(x)| \le \lim_n ||f_n||_\infty$. Since this holds for a.e. $x$, $||\lim_n f_n||_\infty \le \lim_n ||f_n||_\infty$.