Proving an inclusion map is continuous

Question

I wish to prove for $p < q < r$, the inclusion map $L^p \cap L^r \rightarrow L^q$ is continuous. There is a solution to this problem here (last paragraph of #3): https://math24.files.wordpress.com/2013/02/ch6-folland.pdf However, I cannot seem to understand their proof. Would anyone be able to break it down? To me it seems to be more of a verification than a proof.