Show that entire function $f$ is a polynomial of degree at most $n$ Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be a entire function. Suppose that there are $M$, $r>0$ and $n\in \mathbb{N}$ such that $\left|f(z)\right|<M\left|z\right|^n$ for all $z \in \mathbb{C}$ with $\left|z\right|\geq r$.
Show that $f$  polynomial of degree at most $n$.
Remark: I tried to follow the proof of Liuville's Theorem but I have complications with the condition  $\left|z\right|\geq r$..

Well, first of all we know that, as $f$ is an entire function, we can write it as a power series around zero:
$$f(z)=\sum\limits_{m\geq 0}a_mz^m,\:\:\:\forall z\in\mathbb{C},$$
where, by Taylor, $a_m=f^{(m)}(0)/m!$.
Now, if we consider the Cauchy's inequality, we have that
$$|f^{(m)}(0)|\leq m!\frac{\sup\limits_{|z|=r}|f(z)|}{r^m}\leq m!\frac{Mr^n}{r^m}=Mm!r^{n-m},\:\:\:\forall m\in\mathbb{N},\: r>0.$$
Note that if $m > n$, then $0\leq f^{(m)}(0)|\to 0$ when $r\to\infty$. So, we know that $a_m=f^{(m)}/m!=0$ for all $m>n$ and
$$f(z)=\sum\limits_{m=0}^na_mz^m,\:\: \forall z\in\mathbb{C}.$$
So we have proved that $f$ is a polynomial of degree at most $n$.
