Monday, October 17, 2016

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

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