Possible Duplicate:
Limit of $L^p$ norm
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 , but $|f|_{\infty}=\infty$ nevertheless. Could this happen? Imagine $f^{-1}(n,n+1)$ has measure $\frac{1}{n^{n}}$, for example. Then $|f|_{p}$ exists for any $p$, but $|f|_{\infty}=\infty$ nevertheless. However, I do not know how to show $f_{p}$ must be monotonely increasing in this case. Could $f_{p}$ be fluctuating while $|f|_{\infty}=\infty$? I have proved that for $r , $|f|_{p}<\max (|f|_{r},|f|_{s})$. But this does not help to show $|f|_{p}$ does not fluctuate.