I was able to prove that if $f\in \bigcap_{p\ge 1}\textit{L}^p([0,1]) \setminus \textit{L}^{\infty}([0,1])$, then $||f||_p$ diverges as a function of $p$, said better: if we set $F(P):=||f||_p \longrightarrow +\infty$ as $p\rightarrow +\infty$.
Somebody suggested me that a nice application of this is: deduce that $\Gamma(p)^{1/p}$ diverges as $p\rightarrow +\infty$. Just to make it easyer if someone isnt sure about it, recall: $$\Gamma(p)=\int_0^{+\infty}x^{p-1}e^{-x}dx \mbox{, when }p>0\mbox.$$
My attempt consisted in making substitutions to make sure we are on $[0,1]$ with the Gamma integral, and such that i could than overestimate the integral with the integral of a function $g$ in the set mentioned at the beninning so that the proof would finish with something like this: $$\Gamma(P)^{1/p}\ge ||g||_p \longrightarrow \infty \mbox{, as } p\rightarrow +\infty\mbox,$$ thanks to the "lemma" at the beginning.
I'm hoping someone could give me some insight on what i could do better.