Problem on $L^p $ space

Question

Prove that if $0<r<p<s\le \inf$, then $L^r\cap L^s\subset L^{p}\subset L^{r}+L^{s}$.

I've tried to prove it with the Interpolation inequality but unfortunately I'm not able to reach the aim.

Answer 1

For $f\in L^r \cap L^s$ define $A=\{\omega : |f(\omega)|\geq 1\}$. Then $\int |f|^p$ $d\mu=\int_A |f|^p$ $d\mu+\int_{A^C}|f|^p$ $d\mu\leq \int_A |f|^s$ $d\mu+\int_{A^C}|f|^r$ $d\mu$ and since $f\in L^r \cap L^s$ both of these terms are less than $\infty$. If $f\in L^p$ and we define $A$ as before we see that $\mathcal{X}_{A}f\in L^r$ and $\mathcal{X}_{A^C}f\in L^s$