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. 

Here's a hint: Use Chebyshev's Inequality to get that
$$
\int\limits_X|f|^p\,dx\geq N^p~|\{x~|~|f(x)|>N\}|
$$
true for every $N\in\mathbb N$ and $p>1$. From this observation it shouldn't be too difficult. Let me know if you would like the full details.
