Using the Uniform Cauchy Criterion theorem

Question

Suppose $(f_n)$ is defined, continuous on $[a,b]$, and differentiable on this open interval. Then $c \in [a,b]$ and $(f_n(c))$ converge $(f'_n)$ and converge uniformly on $(a,b)$ respectively.

How can I prove that $(f_n)$ converges uniformly on $[a,b]$

I know by uniform cauchy criterion, $(f_n)$ converges uniformly on this set if we can find some $|f_n(x) - f_m(x)| \leq \epsilon$ for all $m, n \geq N$ and all $x \in a$

But we are only given that there is a single point $f_n(c)$ where it is uniformly convergent, nothing about the entire interval $[a,b]$

Answer 1

Hint:

$$f_n(x) - f_m(x) = [(f_n - f_m)(x) - (f_n - f_m)(c)]+(f_n - f_m)(c).$$

Use the MVT on the difference in brackets.