Problem: If $f_n \to f$ in the $L^1$ norm, and $f_n \in L^1$ for each $n$, show that for every $\epsilon > 0$, there exists $\delta > 0$ such that if $m(A) < \delta$, $\int_A |f_n| < \epsilon$ for all integers $n$.
My attempt: We have already shown that if $f$ is integrable, then for each $\epsilon > 0$ there is a $\delta > 0$ such that whenever $m(A) < \delta$, $\int_A f(x) dx < \epsilon$.
To begin with, I noted that $L^1(R)$ is complete, and therefore $f \in L^1(R)$. Now, it's clear that for any $\epsilon > 0$, I can find a $\delta > 0$ such that for some $N$, we have $\int_A |f_n| < \epsilon$ if $m(A) < \delta$ and $n \ge N$. But I can't figure out how to generalize this to all $n$! A hint in the correct direction would be much appreciated.
Note: I have already tried using that
$$\int_A|f_n| \le \int_A |f_n - f| + \int_A |f|,$$
but this first estimate is where I'm running into problems, since it's only bounded by $\epsilon$ for $n \ge N$, where $N$ is some integer.