Let $f\in C^\infty$
$\exists L>0: \forall x\in\mathbb{R}$ and $\forall n\geq 1$ $$|f^{(n)}(x)| \le L,$$
Let $$f(0)=0$$ Prove that $$f(x)=0$$
I using Taylor series but the rest remains and do not know how to eliminate it.
Taylor: $$f(x)=f(0)+f'(0)(x-0)+\frac {f''(0)}{2}(x-0)^2+o_2(x-o)$$
This is a false result.
Take for instance
$$f\colon x\mapsto \sin(x).$$
You have:
$f\in C^\infty(\mathbb R)$,
$\vert f^{(n)}\vert\leqslant 1$,
$f(0)=0$.
But obviously $f(x)\ne 0$ if $x\notin \pi\mathbb Z$.