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

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

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