Taylor Series of Analytic Function has infinite Radius of Convergence

Theorem
Let $F$ be a complex function.

Let $F$ be analytic everywhere.

Let the restriction of $F$ to $\R \to \C$ be a real function $f$.

This means:
 * $\forall x \in \R: \map f x = \map \Re {\map F {x, 0} }, 0 = \map \Im {\map F {x, 0} }$

where $\tuple {x, 0}$ denotes the complex number with real part $x$ and imaginary part $0$.

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