Let $k \in \Bbb N_0, n \in \Bbb N, D \in \Bbb R^n, f:D \to \Bbb R,x \in \overset{\circ}{D} , \alpha \in \Bbb N_0^n$ and $c_{\alpha}, \overline{c}_{\alpha} \in \Bbb R$ with $$\lim_{\lambda \to 0} \frac{f(x+\lambda)- \sum_{|\alpha| \leq k} c_{\alpha} \lambda^{\alpha}}{| | \lambda| |^k} = 0$$ and $$\lim_{\lambda \to 0} \frac{f(x+\lambda)- \sum_{|\alpha| \leq k} \overline{c}_{\alpha} \lambda^{\alpha}}{| | \lambda| |^k} = 0$$ Show that $c_{\alpha} = \overline{c}_{\alpha}$ for all $\alpha \in \Bbb N_0^n$ with $| \alpha| \leq k$. I reduced it to $$\lim_{\lambda \to 0} \frac{\sum_{|\alpha| \leq k} (\overline{c}_{\alpha}-c_{\alpha}) \prod_{j =1}^n \lambda_j^{\alpha_j}}{| | \lambda| |^k} =0$$ but idk how to continue solving the problem.
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