Help show $\lim \limits_{p \to \infty }\frac{{\int_0^1 {|f{|^{p+1}}dx} }}{{\int_0^1 {|f{|^p}dx} }}$ if $f \in {L^\infty }[0,1]$
Show $\lim \limits_{n \to \infty} \frac{a_{n+1}}{a_n} = \|f\|_{\infty}$ for $f \in L^{\infty}$