Ascoli-Arzelà example and equicontinuity My teacher did an example of Ascoli-Arzelà Theorem that I really don't understand, above all how he show that a series of functions is equicontinuous.
Let $f_n:=x^n \quad , \quad x\in [0,1] \quad , \quad f_n\subset C([0,1],\Bbb{R})$
$\forall x, \quad \exists \lim_{x\to +\infty}f_n(x)=f(x) \quad  with \quad f(x):= \begin{cases}
0  & \text{if $0\le x<1$} \\
1 & \text{if $x=1$}
\end{cases}$
$f(x)$ is not continuous $\to$ $f_n$ doesn't uniformly converge to $f$
Now: $f_n(0)=0 \quad \forall n$
And $f_n$ isn't uniformly equi-continuous: $\quad$ $f_n'(x)=(x^n)'=nx^{n-1} \qquad f_n'(1)=n \longrightarrow +\infty$
I really don't understand this last part: Why did he calculated the derivate in $1$? And why can he say that $f_n$ isn't uniformly equi-continuous in this way?
And does a correlation exist between derivates and equicontinuity?

I don't think the behavior of the derivative at the point $x=1$ proves (without much effort) that the sequence is not uniformly equi-continuous . So your teacher's argument is not complete and not a good approach for this.
If the sequence is uniformly equicontinuous then there exists $\delta >0$ such that $|f_n(x)-f_n(y)| <1-\frac 2  e$ for all $n$ whenever $|x-y| \leq \delta$. In particular $|f_n(1-\frac  1 n)-f_n(1)| <1$ for all $n$ sufficiently large. This lead to the contradiction $1-\frac 1 e < 1-\frac  2 e$
