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Proof

(related to Corollary: Taylor's Formula for Polynomials)

• By hypothesis, $I\subset\mathbb R$ is an interval, $f:I\to\mathbb R$ is a $(n+1)$ times continuously differentiable function with $f^{\{n+1\}}(x)=0$ for all $x\in I.$
• In the Taylor's formula, the remainder term vanishes $$R_{n+1}(x)=\frac 1{n!}\int_a^x (x-t)^nf^{\{n+1\}}(t)dt=0$$ and we have $$f(x)=f(a)+\sum_{k=1}^n \frac{f^{\{k\}}(a)}{k!}(x-a)^k$$ for all $a,x\in I.$
• This means with the formula of $n$th power, that $f(x)$ is a real polynomial of degree $\le n.$
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References

Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983