Let f be infinitely differentiable. Suppose $|f(x)| \leq c_n |x|^{n}$ in some neighborhood of 0, for each n. Show that $f^{(n)} (0)=0$ for all n.

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Let f be infinitely differentiable. Suppose $|f(x)| \leq c_n |x|^{n}$ in some neighborhood of 0, for each n. Show that $f^{(n)} (0)=0$ for all n.

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