How does one visualize sequence of derivatives and functions in a same diagram?

Question

Currently I am studying one theorem in real analysis, which states that

Suppose $(f_n)$ is sequence of differentiable functions on $[a,b]$ and $(f_n')$ converges uniformly on $[a,b]$. If there exists $x_0 \in [a,b]$ such that $\lim_{n \rightarrow \infty}{f_n(x_0)}$ exists, then there exists a function $f$ such that $(f_n)$ converges uniformly to $f$ on $[a,b]$ and $f'=\lim_{n \rightarrow \infty}{f_n'}$.

The main idea of the theorem is that if sequence of derivative functions converges uniformly, then the sequence of original functions also converges uniformly.

Question: How to understand this theorem in a picture? I try to visualize both $(f_n')$ and $(f_n)$ in a same picture but to no avail.

Answer 1

You can use Desmos.com to graph some pretty neat things, for example, see here. All you have to do is set some function $f_n(x)$ and some function $g_n(x)=\frac d{dx}f_n(x)$ and have a slider bar for $n$ that goes from $n=1$ to whatever value you want to reach. It can then animate the transitions as $n\to\infty$.

Answer 2

Without the hypothesis "There exists an $x_{0} \in [a, b]$ such that $\lim f_{n}(x_{0})$ exists", you could have $f_{n} \equiv n$ on $[a, b]$; the derivative sequence is identically zero (so the derivatives converge uniformly to $0$) but the sequence $(f_{n})$ converges nowhere.