Lebesgue integral of L^1 function is differentiable

Let $f\in L^1(\mathbb{R})$, and define the function

$$F(x)=\int_a^xf(t)dt.$$

I want to prove that $F$ is almost everywhere differentiable and that $F'(x)=f(x)$ where $F$ is differentiable.

I am following this text.

First one suppose that $F$ is bounded. The author says to define$$g_n(x)=\frac{F(x+1/n)-F(x)}{1/n},$$and that $\lim_{n\to\infty}g_n(x)=F'(x)$ for almost every $x$.

I'm ok that IF THE LIMIT EXISTS, its value is $F'(x)$.

QUESTION : How do I prove that the limit exists for almost every $x$ ?

This question is related to this one, but here I am asking for a proof.