Prove that $\cap_{0<p<\infty}L^p([0,1])\supset L^{\infty}([0,1])$.
My work-
let $f \in L^{\infty}([0,1])$. then $\vert f\vert_{\infty}<M\ for some M\ a.e \ on [0,1]$.
we know that $|f|<|f|_{\infty}<M\ a.e$.
then, $|f|^p<M^p\ a.e $
therefore, $\int|f|^p<\int M^p\ which\ is \ finite $.
this implies $f\in \cap_{0<p<\infty}L^p([0,1])$.
hence $\cap_{0<p<\infty}L^p([0,1])\supset L^{\infty}([0,1])$.
Is this correct? Am I missing something?