Derivative of Uniformly Convergent Sequence of Differentiable Functions

Theorem
Let $J$ be a bounded interval.

Let $\left\langle{f_n}\right\rangle$ be a sequence of real functions $f_n: J \to \R$.

Let each of $\left\langle{f_n}\right\rangle$ be differentiable on $J$.

Let $\left\langle{ f_n \left({ x_0 }\right) }\right\rangle$ be convergent for some $x_0 \in J$.

Let the sequence of derivatives $\left\langle{f_n'}\right\rangle$ converge uniformly on $J$ to a function $g : J \to \R$.

Then $\left\langle{f_n}\right\rangle$ converge uniformly on $J$ to a differentiable function $f: J \to \R$ and $D_x f = g$.