This is the proof of the fundamental theorem of algebra (FTA) given in Hatcher's Algebraic Topology textbook (I have underlined the relevant part):
Could someone explain why $r$ needs to be bigger than $1$ for the proof to work?
I don't see where the assumption $r>1$ is used in the proof. I am sure I am missing something obvious...
It's for the inequality $$ (|a_1|+\cdots+|a_n|)|z^{n-1}|\geq |a_1z^{n-1}+\cdots+a_n| $$ i.e. $$ |a_1z^{n-1}+\cdots+a_n|\leq|a_1||z|^{n-1}+\cdots+|a_n|\leq |a_1||z|^{n-1}+\cdots+|a_n||z|^{n-1}. $$