Using only Proposition (1) and a compactness argument, WITHOUT The fundamental theorem of algebra, show that there is a point $z_{0}$ such that $$\left | p(z_{0}) \right | = \inf \left \{ \left | p(z) \right | : z \in \mathbb{C} \right \} $$.
Proposition (1):
$$ p(z) = a_{0} + a_{1}z + ... +a_{n}z^{n}$$ with $a_{n} \neq 0 $. Then $$ \frac{p(z)}{a_{n}z^{n}} \rightarrow 1 $$ for $z\rightarrow \infty$
I would appreciate any help!