If $f\not\in L^{\infty}$ then $\lim_{p\rightarrow\infty}||f||_p=\infty $. I want to prove that

If $f\not\in L^{\infty}$ then $\lim_{p\rightarrow\infty}||f||_p=\infty $.

I'd like any hint that approaches to solution.
I now that for every real number $r$ the set $\{x:|f(x)|>r\}$ has positive measure $f$ but I do not know what to do next. 

If $f \not\in L^{\infty}$, then given any $M > 0$, we have $|f| > M$ on a set $E$ of positive measure. Therefore, $|f|^p > M^p$ on $E$, so
$$\int |f|^p \geq \int_E |f|^p \geq M^p \mu(E)$$
If $\mu(E) = \infty$ then this shows that $\|f\|_p = \infty$ for all $p<\infty$, so the result certainly holds.
Otherwise,
$$\|f\|_p = \left(\int |f|^p\right)^{1/p} \geq M \mu(E)^{1/p}$$
As $0 < \mu(E) < \infty$, we have $\lim_{p \to \infty}\mu(E)^{1/p} = 1$, so
$$\liminf_{p \to \infty}\|f\|_p \geq M$$
As this holds for arbitrarily large $M$, we conclude that $\lim_{p \to \infty}\|f\|_p = \infty$.
