A real-analytic function at $p$ is $C^\infty$ at $p$ (in “An Introduction to Manifolds” by Loring W. Tu.)

Question

I am reading "An Introduction to Manifolds" by Loring W. Tu.
There is the following sentence in this book.
I cannot understand what the sentence is saying.

A real-analytic function is necessarily $C^\infty$, because as one learns in real analysis, a convergent power series can be differentiated term by term in its region of convergence.

By definition we can expand a function which is real-analytic at $p$ in its Taylor series at $p$.
So obviously a function which is real-analytic at $p$ must be $C^\infty$ at $p$.

I think we don't need "because as one learns in real analysis, a convergent power series can be differentiated term by term in its region of convergence".

I cannot understand what Loring W. Tu is saying.

Please explain.

Answer 1

You are confusing

forall $K$, $f(x)= \sum_{|a|\le K} c_a x^a+O(|x|^{K+1})$

with

forall $K$ and for all $y$ close to $0$, $f(x)= \sum_{|a|\le K} c_a(y) (x-y)^a+O(|x-y|^{K+1})$ and the $c_a(y)$ are continuous

$C^\infty$ at (near) $0$ is the latter.

That a (convergent) power series is $C^\infty$ follows from that (for $|x-y|$ small enough) $$\sum_a c_a x^a= \sum_a c_a (y+(x-y))^a=\sum_a c_a \sum_b {a\choose b}y^a (x-y)^b=\sum_b (x-y)^b\sum_a c_a {a\choose b}y^a$$ is a convergent power series in $x-y$