# calculation – Question about infinitely differentiable functions

Leave $$f: (- 1,1) rightarrow mathbb R$$ be infinitely differentiable in $$(- 1,1)$$ with $$f (0) = 1$$ and has the following properties:

(one) $$vert f ^ {(n)} (x) vert le n!$$ for each $$x in (-1, 1)$$ and for each $$n en mathbb N$$

(two) $$f & # 39; ( frac {1} {m + 1}) = 0$$ for each $$m en mathbb N$$

I need:

Find the value of $$f ^ {(n)} (0)$$ for each $$n en mathbb N$$

Determine the value of $$f (x)$$ for each $$x in (-1,1)$$

I affirm that $$f ^ {(n)} (0) = 0$$ for all $$n en mathbb N$$ And I wanted to prove this by induction. When $$n = 1$$, we noticed that by (2), $$f & # 39; (0) = 0$$ by the sequential criterion of limits with $$m rightarrow infty$$. However, I'm not sure how to prove the inductive step.

I note that since (1), we have $$f (x) le frac {1} {1-x}$$ by Taylor's theorem.

Any help is really appreciated!