I have tried to prove the following theorem by following the proof in my lectures. My question is similar to this, but at a lower level (I do not know about Banach spaces). May I ask if my understanding is correct? Thank you.
Theorem. Let $(f_n)_{n∈\mathbb{N}}$ be a sequence of continuously differentiable functions on $[a, b].$ Assume that (i) $(f′_n)_{n∈\mathbb{N}}$ converges uniformly to a function $g : [a, b] \to \mathbb{R}$; (ii) $(f_n(a))_{n∈N}$ converges in $\mathbb{R}$. Then $(f_n)_{n∈N}$ converges uniformly to a function $f$ on $[a, b]$ that is continuously differentiable and satisfies $f′ = g$ on $[a, b]$.
By the corollary to the Fundamental Theorem of Calculus (FTC), we have that $$f_n(x) = f_n(a) + \int_a^x f_n'(t) dt.$$ for each $x ∈ [a, b]$. Note that $g$ is continuous on $[a, b]$, by Weierstrass' theorem. We claim that $f$ is defined by $$f(x) = L + \int_a^x g(t) dt,$$ for each $x \in [a,b]$ and where $L := \textrm{lim}_{n \rightarrow \infty} f_n(a).$ By the FTC, $\int_a^x g(t) dt$ is differentiable on $[a, b]$ and its derivative is equal to $g$ on $[a,b]$. By the sum rule for derivatives, $f'(x) = 0 + g(x) = g(x)$ for each $x \in [a,b]$. As g is continuous, f is continuously differentiable. It now remains to show that the convergence is uniform. Note that, by subtracting the above equations and using the linearity of the integral, we obtain: $$ |f(x) - f_n(x)| = |L - f_n(a) + \int_a^x (g-f'_n)(t) dt|.$$ By the triangle inequality and the continuity of the integral, we obtain: $$ |f(x) - f_n(x)| \leq |L - f_n(a)| + ||g-f_n||_{[a,x]}(x-a) \leq |L - f_n(a)| + ||g-f'_n||_{[a,b]}(b-a).$$ By hypothesis, since $f_n(a) \to L$ and $f' \to g$ uniformly, we obtain that $$ f_n \to f$$ uniformly on [a,b].
Is there any parts of the proof which are incorrect/can be simplified? Thank you.