Let $n \geq 1$ be an integer, and $C: \mathbb{N}^n \to \mathbb{R}$ be a function. Prove that there exits a infinitely differentiable function $f: \mathbb{R}^n \to \mathbb{R}$ whose value and partial derivatives of all orders at $0 \in R^n$ satisfy $$\left( {\partial^{|\beta|}f \over{\partial x_{1}^{\beta_{1}}\partial x_{2}^{{\beta }_{2}}}...\partial x_{n}^{{\beta}_{n}}} \right) (0)=C(\beta),$$for all $\beta\in \mathbb{N}^n.$

I know a borel theorem in one-variable case,but I am not sure whether they have some correlations. Can someone help me prove thia question?


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