The general proof of the pb 7.2 (a) is in page 126 Wheeden and Zygmund.
The proof shows that for every finite measure space, $ ||f||_{n} \arrow ||f||_{\infty} $.
For (b),
$ lim_{n \arrow \infty} \frac{int_{X}|f|^{n+1}d \mu}{int_{X}|f|^{n}d\mu} = lim_{n \arrow \infty} \frac{||f||_{n+1}^{n+1}}{||f||_{n}^{n}}=lim_{n \arrow \infty} (\frac{||f||_{n+1}}{||f||_{n}})^{n}||f||_{n+1}=||f||_{\infty} $
