I will put a partial answer to my question by combing Davide's proof and my previous work.
Suppose $|f|_{\infty}=a$, then $(r,a)\in E$ and by the continuity of $\phi$ we proved the statement. So we need to prove this under the hypothesis $|f|_{\infty}=\infty$. We have two cases:
(1) $|f|_{p}<\infty,\forall p<\infty$. Then we need to show $|f|_{p}\rightarrow \infty$ as $p\rightarrow \infty$.
(2) There exist some $p>r$ such that $|f|_{p}=\infty$. We need to show $\forall q>p$, $|f|_{q}=\infty$ as well.
Now (2) is straightforward since if $|f|_{q}<\infty$, then $p\in [r,q]$ implies $|f|_{p}<\infty$ as well. This contradicts with our hypothesis. So it suffice to prove (1). But Davide already showed via a clever argument that $\lim \inf |f|_{p}\ge |f|_{\infty}$. So this force $\lim \inf |f|_{p}=\infty$ as well.