if f is integer and satisfies $| f (z) | ≤ | z |^n$ then f must be a polynomial

Question

Show that if f is integer and satisfies an inequality of the form $| f (z) | ≤ | z |^n$ for some $n ∈ \mathbb N$ and for all $| z |$ big enough, then f must be a polynomial.

I know that since f is integer it's equal to a power series centered at zero$$f(z)=\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}z^n$$ but I don't know how to proceed

Answer 1

Hint: try to use the Cauchy integral formula for the $n$-th derivative $$ f^{(n)}(0)=\frac{n!}{2\pi i}\oint_{|\zeta|=R}\frac{f(\zeta)}{\zeta^{n+1}}d\zeta $$ for sufficiently large $R$.