Limit of consecutive Lp norms

Limit of consecutive Lp norms

I've been wrestling with the following proof off and on for a number of
days, and I'm in need of a nudge in the right direction.
Let $(E,\mathcal{M},\mu)$ be a measure space with $0 < \mu(E) < \infty$.
Consider $f \in L^\infty(E)$ with $\|f\|_\infty > 0$; show that
$$ \lim_{n\to\infty} \|f||_n = \lim_{n\to\infty}
\frac{\|f\|_{n+1}^{n+1}}{\|f\|_n^n} = \|f\|_\infty $$
Now I'm familiar with the result and proof that under these circumstances
$\lim_{n\to\infty} \|f||_n = \|f\|_\infty$, so I've been focusing on
somehow showing the two limits to be the same.