I'm trying to understand if it's true that: " if $f\in L^p\quad \forall p\in N\implies f\in L^\infty$"? My thoughts: Since $\int_R |f(x)|^p dx<\infty\quad\forall p\implies |f(x)|\to 0$? Can anyone help me please? Thank you.
2026-04-28 15:15:25.1777389325
$L^\infty$ and the intersection of the spaces $L^p$
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Hint: Consider $$ f(x)=\left\{\begin{array}{} -\log(x)&\text{if }x\in(0,1)\\ 0&\text{otherwise} \end{array}\right. $$