Taylor Series of Analytic Function has infinite Radius of Convergence

Theorem
Let $F$ be a complex function.

Let $F$ be analytic everywhere.

Let $f = F {\restriction_{\R}}$ be a real function.

Let $x_0$ be a point in $\R$.

Then:
 * the Taylor series of $f$ about $x_0$ converges to $f$ at every point in $\R$

Proof
The result follows by Convergence of Taylor Series of Function Analytic on Disk for the case $R = \infty$.

Also see

 * Taylor Series reaches closest Singularity