Ring of germs and induced isomorphism Let $A$ be the ring of germs of real analytic functions in $0\in\mathbb R$. Let $x\in A$ be the identity map on $\mathbb R$. How can I show that the map $f\mapsto \sum_n(f^{(n)}(0)/n!)T^n$ from $A$ to $\mathbb R[[T]]$ is injective, and induces an isomorphism from the $xA$-adic completition of $A$ onto $\mathbb R[[T]]$? And why this is false if we replace $A$ by $C^\infty$-functions in $0$? Qing Liu: Algebraic Geometry and Arithmetic Curves problem 1.3.3.

For the first question: the map is injective simply because the kernel is the set of germs of analytic functions all of whose derivatives at $0$ are zero, and there is clearly only one such germ. To show that the map is surjective on the completion it is enough to show that it is surjective modulo powers of the maximal ideals. Can you do that?
For the second question: There is a famous theorem of Borel that says that every formal series in $\mathbb R[[T]]$ is the Taylor series of a $C^\infty$ function. It follows easily from this that in the $C^\infty$ case the map is not surjective.
