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 $[a,b]$