Let $f:[a,b]\rightarrow \mathbb R$. We say that $f$ is $\mathbb R$-analytic if for each $x_0 \in [a,b]$ there is $R(x_0)>0$ and power series $\sum_{k=0}^\infty c_k(x_0)(x-x_0)^k$ convergent for $x\in (x_0-R(x_0), x_0+R(x_0))$ such that $f(x)=\sum_{k=0}^\infty c_k(x_0)(x-x_0)^k$ for $x\in [a,b] \cap (x_0-R(x_0), x_0+R(x_0))$.
I wish to show elementary, without the Cauchy formula and complex analysis's theorems, that if $f:[a,b]\rightarrow \mathbb R$ is analytic then there is $R>0$, non depending on $x_0$, such that for each $x_0 \in [a,b]$ there is a power series $\sum_{k=0}^\infty c_k(x_0)(x-x_0)^k$ converegent for $x\in (x_0-R, x_0+R)$ such that $f(x)=\sum_{k=0}^\infty c_k(x_0)(x-x_0)^k$ for $x\in [a,b] \cap (x_0-R, x_0+R)$.