If $\lim_{n→∞} f_n = f$ pointwise on an interval $I$ and if each $f_n$ is twice differentiable on $I$, then $\lim_{n→∞} f_n = f$ uniformly on $I$? Is this true?
We can say this is true if there is a function $g$ such that $\lim_{n→∞} f_n' = g$ uniformly on $[a,b]$ and,
the is a point $c∈[a,b]$ for which the limit $\lim_{n→∞} f_n(c)$ exists.
How do we prove this?