$L^1$ convergence for product of arbitrary $L^1$ function with sequence of bounded Lebesgue measurable functions with zero integral on subintervals

I'm studying for a qualifying exam, and I came across this question that I wasn't able to solve:

Let $(f_n)_{n \in \mathbb{N}}$ be a sequence of Lebesgue measurable functions defined on $[0, 1]$ such that $|f_n(x)| \le C$ for a.e. $x \in[0, 1]$ and every $n$. Further assume that $\lim_{n \to \infty} \int_0^a f_n(x) dx = 0$ for every $a \in (0, 1)$. Prove that for any given $g \in L^1([0, 1])$,

$$\lim_{n \to \infty} \int_0^1 g(x) f_n(x) dx = 0.$$

I'm sure I'm missing something simple, so I am happy to receive any small hints. I have tried several methods. First, if $f_n \to 0$ a.e., then the result would follow by LDCT, but this is not true in general, as I have found a counterexample. Similarly, I thought about using Holder's inequality and the fact that $f_n \in L^\infty([0, 1])$ for every $n$, but I'm having trouble controlling terms of the form $\int_0^a |g(x) f_n(x)| dx$ as Holder's inequality doesn't seem to help as $g \notin L^\infty([0, 1])$ in general.