$\{p \in [1, \infty]: f \in L^p\}$ is an interval?

Question

I understood the proof but why does this imply that $\{p \in [1, \infty]: f \in L^p\}$ is an interval?

Answer 1

The inequality implies that if $f\in L^p$ and $f\in L^q$ for $p<q$, then $f\in L^r$ for all $p<r<q$. This implies that the set is an interval.