Uniformly bounded derivatives implies globally analytic

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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 0.