I am styding a bit of $\mathcal{L}_p$ spaces and, in particular, where $p=\infty$ and it is already clear to me that $\lim_{\rightarrow \infty} \|f\|_p = \|f\|_\infty$, but I've met with the following exercise, that I could not prove:
Let $\mu$ be a finite measure in $(\Omega, \mathcal{F})$ and $f:\Omega\rightarrow \mathbb{R}$ be a measurable function with $0<\|f\|_\infty<\infty$. Then, $\lim_{n\rightarrow \infty} \frac{\int_\Omega |f|^{n+1}d\mu}{\int_\Omega |f|^n d\mu} = \|f\|_\infty$
Could anyone give me a hand?