I'm trying to show that if $f$ is infinitely differentiable everywhere (on $\mathbb{R}$) and there exists $C>0$ such that for each $n\in \mathbb{N}$ $$|f^{(n)}(x)|<\frac{C(n!)}{|x-x_0|^n}$$ then $$f(x)=\sum_{n=0}^{\infty } \frac{f^{(n)}(x_0) }{n!}(x-x_0)^n $$ for all $x\in \mathbb{R} .$
I think I need to show that by Taylor's theorem that the remainder $$|\frac{f^{(n+1)}(k)(x-x_0)^{n+1}}{(n+1)!}|\rightarrow 0 , \ k \text{ between } x_0 \text{ and } x,$$ but I'm unsure on how to show this.