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.