$a < b$ are real numbers and the function $f : (a, b) \to \mathbb{R}$ is analytic.
Show that for each $x_0 ∈ (a, b)$ there exist $c \in (a, x_0)$ and $d \in (x_0, b)$ and positive constants $c_0$, $d_0$ such that for each $x \in (c, d)$ and each $n \in \mathbb{N}$ there exists:
$$\frac{|f^{(n)}(x)|}{n!} \leq c_0d_0^n$$
It feels super complicated and I don't even know how to approach that proof. Here it is is in the opposite direction: Smooth Function with Derivative Nonnegative is Analytic
In the book where the exercise comes from there is a hint that:
$$\sum_{k = n}^{\infty} \frac{k!}{(k-n)!} x^{k-n} = \frac{n!}{(1-n)^{n+1}}$$