real analysis - How to show $|f|_{p}rightarrow |f|_{infty}$?

I was asked to show:

Assume $|f|_{r}<\infty$ for some $r<\infty$. Prove that $$

|f|_{p}\rightarrow |f|_{\infty}
$$ as $p\rightarrow \infty$.

I am stuck in the situation that $|f|_{p}<\infty$ for all $r

Could $f_{p}$ be fluctuating while $|f|_{\infty}=\infty$? I have proved that for $r