Image of non-constant entire function is dense in $\mathbb{C}$

Question

how to show image of a non constant entire function is dense in $\mathbb{C}$?

Here answer is given.But I have a doubt.

In the proof using Casorati-Weierstrass Theorem, it is proved that f can have only finitely many terms.

Thus $f=a_0+a_z+...+a_n z^n.$

Then Fundamental theorem of Algebra guarantees that f have n roots in $\mathbb{C}$, But how can we say f is constant?

Answer 1

At that point in the proof it has been established that $|f(z)-\alpha|>0$ for all $z$. So the polynomial $p(z) = f(z)-\alpha$ has no roots. If it is non-constant you have a contradiction to the fundamental theorem of Algebra.