There is one step in the proof the following proof that I would love to understand the intuition.
Theorem 14.7.1.
Let $[a,b]$ be an interval, and for every integer $n \geq 1$, let $f_n : [a,b] \rightarrow \mathbb{R}$ be a differentiable function whose derivative $f_n^{\prime} : [a,b] \rightarrow \mathbb{R}$ is continuous. Suppose that the derivatives $f_n^{\prime}$ converge uniformly to a function $g : [a,b] \rightarrow \mathbb{R}$. Suppose also there exists a point $x_0$ such that the limit $\lim_{x\to\infty} f_n(x_0)$ exists. Then the functions $f_n$ converge uniformly to a differentiable function f, and the derivative of f equals g.
I don't understand the intuition of how g is defined below. More specifically. What I don't understand is why we have the term $L - \int_{[a,x_0]} g$ in the proof below. I have checked that it does work, but I don't understand why it works.
Proof:
