Question on $L^p$ spaces

Question

Let $f:[0,1] \to \mathbb{R}$ be a measurable function. Prove that $$ f \in L^\infty([0,1];\mathbb{R}) \iff f \in L^p([0,1];\mathbb{R}) \ \ \forall p \ge 1 \ \text{ and } \sup_{p\ge 1}\|f\|_p<\infty. $$ Remark: I've already solve the first part, i.e. if $f \in L^\infty([0,1];\mathbb{R})$ then ...

Answer 1

Define for $n$ integer $E_n:=\{x\in [0,1],g(x)\geqslant n\}$, where $g$ is a measurable function in the class of $f$ for equality almost everywhere.

Then $|g(x)|^p\chi_{E_n}(x)\geqslant n^p\mu(E_n)$, which gives, integrating that $$n\mu(E_n)^{1/p}\leqslant \sup_{p\geqslant 1}\lVert g\rVert_p=:M<\infty.$$ So $\mu(E_n)\leqslant \left(\frac Mn\right)^p$. For $n\geqslant M+1$, this gives that for any $p>1$, $$\mu(E_n)\leqslant \left(\frac M{M+1}\right)^p,$$ and taking the limit $\lim_{p\to +\infty}$, we get that $\mu(E_{M+1})=0$, hence $f\in L^\infty$.