Let $a>0$ and the sequence $f_n:[-a,a] \rightarrow \mathbb{R}$. Suppose that the following conditions are true:
$\bullet$ There is $k \in \mathbb{N}\cup\{0\}$ such that $f_n$ is $(k+1)$-differentiable in $[-a,a]$;
$\bullet$ $f_n^{(i)}(0)=0 \ \forall \ i \in \{0,1,\dots,k\}$
$\bullet$ $|f_n^{k+1}(x)|\leq C.$
Prove that there is a subsequence in $(f_n)_{n \in \mathbb{N}\cup \{0\}}$ that is uniformly convergent in $[-a,a]$.
I know that this must be like this question Ascoli's Theorem, but i'm having some hard time one this.
Hint: if $f$ is differentiable and $|f'| \leq C$, then $|f(x)-f(y)| \leq C|x-y|$. Moreover, $|f(x)| = |f(x)-0| \leq C|x-0| \leq Ca$.
Apply this to $f_n^k$ for each $n$. The bound on their derivatives is the same for all of them, so the Lipschitz constant you get is the same for all. The proceed likewise unitl you reach each $f_n$, keeping track of how the bound changes. It will be uniform; this will allow you to conclude that $\{f_n\}$ is uniformly bounded and uniformly equicontinuous. Then apply Arzelà-Ascoli.