Proof of Borel's lemma on derivatives and real sequences

[Castilla]

There is a theorem (the "Borel lemma") that says: Let (A_n) by any sequence of real numbers. We can built a function "F", indefinitely differentiable, such that if G is the n-derivative of f, G(0) = a_n.

Does someone knows a proof or where can I find it? The theorem appears in wikipedia, under the name "Borel Lemma", but all the wiki information is that this theorem is sometimes useful in PDE...

 

[HallsofIvy]

At first glance I thought you were talking about Taylor's series. But Borel's Lemma is more general than that: it says that if have a sequence of smooth, complex valued functions, f_n(x), on an open subset of Rⁿ, then there exist F(x,t) such that
$$\frac{\partial^k f}{\partial t^k}= f^k(0, x)$$.

 

[Castilla]

Yes, that is the theorem. Only that the version in my book is simplified from complex to real numbers and from partial derivatives to simple derivatives.

Sadly, I can't follow all the steps in my book, that is why I ask if you know some source where I can find the proof.