# A problem on complete metric space

Prove: If $f\in C^{\infty}[a,b]$, and $\forall x\in[a,b]$, $\exists n\in \mathbb{N}$, $\forall m>n$, $f^{(m)}(x)=0$, then $f$ is a polynomial.

It's a problem I got from Zorich's Mathematical Analysis Vol.II, Chap 9.5 Prob. 2(b).

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Does $n$ depends on $x$? –  John Ma Nov 20 '13 at 7:24
Yes, for different $x$ there is different $n$. –  Andy Nov 20 '13 at 7:30
Perhaps overkill, but this can be done using the Baire Category theorem (with weaker assumptions, I'd add). –  user61527 Nov 20 '13 at 7:33
@T.Bongers: From the title might be it is not an overkill? ^^ –  John Ma Nov 20 '13 at 7:34
Thanks a lot! This is exactly the solution I am looking for! –  Andy Nov 20 '13 at 7:42