I am trying to prove a theorem, and I have been able to reduce it to the following question. I feel that this should be easy, but I can't see the solution.
If $(g_n)_{n\geq 1}$ is a sequence of functions in $L^1$ such that $\lim_{n\to\infty} \|g_n\|_1 = 0$, then there exists an $N$ such that $g_N$ is in $L^2$. Here, $\|\cdot\|_1$ is the norm in $L^1$.
I am not even sure this is true, but any help would be appreciated.
The claim is actually false in $[0,1]$ with Lebesgue measure.
A counterexample can be seen by consider the functions $f_n = \frac{1}{n\sqrt{x}}$, then the integral is $\frac{2}{n}$ so each are in $L^1$ and $\left|\left|f_n\right|\right|_1 \rightarrow 0$.
However, for all $n$ we have $f_n^2 = \frac{1}{n^2x}$, so $f_n \notin L^2$ for any $n$.