Let $(f_{n})_{n}$ be a sequence of differentiable functions defined on an interval $I \subseteq \mathbb{R}$ with values in $\mathbb{R}$. Suppose that $(f'_{n})_{n}$ converges uniformly on $I$ and that there exists an $x_{0} \in I$ for which $(f_{n}(x_{0}))_{n}$ converges. Show that $(f_{n}(x))_{n}$ converges for every $x \in I$.
My attempt: We have to prove that the sequence converges pointwise, but we do not 'know' the limit function. Therefore, I chose to work with a Cauchy sequence. Take an $\epsilon >0$ and an $x \in I$. Then will $\vert f_{n}(x)-f_{m}(x) \vert \leq \vert f_{n}(x)-f_{n}(x_{0}) \vert + \vert f_{n}(x_{0})-f_{m}(x_{0}) \vert + \vert f_{m}(x_{0})-f_{m}(x) \vert$. We can get the second term on the right hand side smaller than $\frac{\epsilon}{3}$ if we take $n,m$ large enough. However, I suppose the other first and third term on the right hand side were a bad choice because I don't how I can make use of them.
Any help? Thank you in advance.
Hint. By the MVT, for any $n,m$, there exist $t_{n,m}$ between $x$ and $x_0$ such that $$\begin{align}|f_{n}(x)-f_{m}(x)| &\leq |(f_{n}(x)-f_{m}(x))-(f_{n}(x_0)-f_{m}(x_0))|+|f_{n}(x_0)-f_{m}(x_0))|\\ &= |f'_{n}(t_{n,m})-f'_{m}(t_{n,m})|\cdot|x-x_0| + |f_{n}(x_0)-f_{m}(x_0))|\\ &\leq \sup_{t\in I} |f'_{n}(t)-f'_{m}(t)|\cdot |x-x_0| + |f_{n}(x_0)-f_{m}(x_0))|.\end{align}$$
P.S. As pointed out by the comments below, my previous hint $$f_n(x)=f_n(x_0)+\int_{x_0}^xf'_n(t)dt$$ is valid under the additional assumption that $f_n'$ is integrable on $I$.