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!