for each $n$ such that $\int |f_n|^3\rightarrow 0$ and $\int |f_n|^5\rightarrow 0$. Prove or give counter example for each of the statement below.

Question

suppose $f_n:R\rightarrow R$ is Lebesgue measurable for each $n$ such that $\int |f_n|^3\rightarrow 0$ and $\int |f_n|^5\rightarrow 0$. Prove or give counter example for each of the statement below.

1. $\int|f_n|^2\rightarrow 0 $
2. $\int|f_n|^4\rightarrow 0$

My Work-

For the first one, Let $f_n(x)=\frac{1}{n}\chi_{[0,n^2]}$ then $|f_n|^2=\frac{1}{n^2}.n^2=1$. Therefore, $\int|f_n|^3$ does not converge to zero. But, $\int|f_n|^3=\frac{1}{n^3}.n^2=\frac{1}{n}$. This goes to zero. similarly, $\int|f_n|^5=\frac{1}{n^5}.n^2=\frac{1}{n^3}$. This also goes to zero.

Foe the second problem, I could not find such a counter example. I feel like, this is true. How to start the proof? Thank you in advance.

Answer 1

Write $$|f_n|^4 = |f_n|^{3/2} \cdot |f_n|^{5/2}$$ and apply the Cauchy-Schwarz inequality.