let $\displaystyle f:(-1,1) \rightarrow R$ be $\displaystyle C^\infty$. Prove that $\displaystyle f$ is real analytic in some neighborhood of 0 if and only if there is a nonempty interval$\displaystyle (- \sigma, \sigma)$ and a constant $\displaystyle M$ such that $\displaystyle |(d/dx)^k f(x)| \leq M^k * k!$ for all $\displaystyle x \in (- \sigma, \sigma)$ and all $\displaystyle k \in \{1,2,...\}$

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