I'm trying to solve the following practice problem from my university:
If $f_n:\mathbb{R}\rightarrow\mathbb{R}$ is differentiable, such that $||f_n||_{L^1}\leq 1$ and $||f_n'||_{L^1}\leq 1$ for all $n\in\mathbb{Z}^+$. Furthermore, for any $\epsilon>0$ there exists $A_\epsilon$ such that $$\sup_n\int_{|x|>A_\epsilon}|f_n(x)|\,dx<\epsilon.$$ Prove that there is a subsequence of $f_n$ that converges in $L^1$.
So far I have noticed that if the $f_n$ has continuous derivative, or at least when we can apply the fundamental theorem of calculus, then the result follows from a simple application of Arzela Ascoli. Alternatively, if one can show that if $f_n$ is pointwise bounded, then we can apply the Helley's selection theorem (as in wikipedia:https://en.wikipedia.org/wiki/Helly%27s_selection_theorem) However, I don't know how to show pointwise boundedness because $f$ is merely assumed differentiable.
Can someone please help me finish this problem? Thank you!!