Proving that $L^p \cap L^{\infty} \supset L^r$ where $1 \leq p <r < \infty$

Question

Suppose that $1 \leq p <r < \infty$.

Prove that $L^p \cap L^{\infty} \supset L^r$

I can prove this theorem for cases other than $\infty$ but not for those with $\infty$ . Could you please guide me on how to proceed in the infinity case ?

Answer 1

This is not true in general. For instance, take your space to be $X=[1,\infty]$ with the standard measure, $r=2$, $p=1$, and $f(x)=\frac{1}{x}$, then $f$ is in $L^2(X)=L^r(X)$, but $f$ is not in $L^1(X)=L^p(X)$.