Why do many books go through "downsizing" arguments to prove the Fundamental Theorem of Algebra? Isn't this the simplest proof?

Theorem.Every non-constant polynomial with complex coefficients has a zero in .

Proof.Let be any polynomial. If for all , is an entire function. Furthermore, if is non-constant, as and is bounded. By Liouville's Theorem, is constant and so , contrary to our assumption.

Yet most books make this so complicated. Why?