# Every real sequence is the derivative sequence of some function

I am looking for the proof of the following theorem:

Let $(a_n)$ be a sequence of real numbers. Then there exists a function $f$ which is infinitely differentiable at 0, and $$\frac{d^nf}{dx^n}(0) = a_n, \ \ \text{for all } n.$$

I would appreciate either a sketch of the proof or an online reference to it. A general case is listed as Borel's lemma in Wikipedia, without proof.

The hard part is when the power series $\sum_n \frac{a_n}{n!}x^n$ has a zero radius of convergence.

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This is a famous theorem of Borel. If you have Hormander's "The analysis of partial differential operators I" it's Theorem 1.2.6 there. He proves the result in any dimension. The basic idea is to write a sum of functions $\sum_n {a_n \over n!}\phi(m_n x)x^n$ where $\phi(x)$ is $1$ near $x = 0$ and is equal to zero outside of a small set containing $0$. If the $m_n$ are chosen carefully then the sum will have the desired properties.