How to show real analyticity without extending to complex plane Suppose we have some $f \in C^\infty(\mathbb{R},\mathbb{R}).$  For example, $$f(x)=(1+x^2)^{-1}.$$  Using complex analysis, we can easily show $f$ is real analytic.  Is there an easy, general method to show this which doesn't use complex analysis?

The most popular general method is to calculate the general term $f^{(n)}(a)$, and if that's possible, and for every $a$, find an interval $[a-h,a+h]$, such that
if 
$$
M_n=\max_{x\in[a-h,a+h]}\lvert \,f^{(n)}(x)\rvert,
$$
then
$$
\limsup_{n\to\infty} \left(\frac{k_n}{n!}\right)^{\!1/n}=L<\infty.
$$
The $M_n$ can be approximated using Taylor expansion remainder.
