The Question
Suppose $f_n$ :$\mathbb R \to \Bbb R$ is lebesgue measurable for each n such that ||$f_n$||$_3 \to 0$ and ||$f_n$||$_5 \to 0$ as $n \to \infty$. prove or give a counterexample for each of the statements below. $\\ \\\\\\\\\\\\\\$
a.$\Vert f_n \Vert _2 \to 0$ $\\\\\\\\\\\\\\\\\\\\\\\\\\\\$
b.$\Vert f_n \Vert _4 \to 0$
I found a counterexample for part a , it took some time, but I believe that $f_n = \chi_{[1,\infty)}\frac {1} {n\sqrt x } $ works I think. But I feel like I am spinning my wheels on part b and need a push in the right direction. Any help would be appreciated, I feel like I am missing something simple.