The question has been asked before here:if $\|f\|_p \le C$ for all $p \ge 1$, then $f \in L^\infty$. but I solved it my way and would like to know if my solution is correct.
Assume towards contradiction that it is not the case. Then $\lambda(A_n)>0$ where $A_n=\{x| f(x)>n\}$ for all $n$. Now consider the function $n\chi_{A_n}$ its Lp norm is $n\lambda(A_n)^{1/p}\leq C$ thus $\lambda(A_n)\leq (C/n)^p$ but now for $n>C$ we take limit $p \to \infty$ to find $\lambda(A_n)=0$ contradiction. and so the claim is proven.
