I was trying to find a function which is in $L^p[0,1]$ for all $p$ but not in $L^{\infty}[0,1]$. I asked my professor, when he said it's $\frac{1}{x \ln (x)}$. However, I am having a hard time to show how is this true. Any help will be appreciated, thanks.
Edit: I was wondering if the property holds for $\frac{1}{x(ln\frac{1}{x})^2}$.