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 $R = \infty$.

$\blacksquare$