Uniformly bounded derivatives implies globally analytic

Statement

Global statement

Suppose $f$ is an infinitely differentiable function on $\R$ such that, for any fixed $a,b \in \R$, there is a constant $C$ (possibly dependent on $a,b$) such that for all nonnegative integers $n$, we have:

$|f^{(n)}(x)| \le C \ \forall x \in [a,b]$

Then, $f$ is a globally analytic function: the Taylor series of $f$ about any point in $\R$ converges to $f$. In particular, the Taylor series of $f$ about 0 converges to $f$.

Examples

The functions $\exp, \sin, \cos$ all fit this description.