Multivariable Advanced Calculus

292 BROUWER DEGREE19. Using Problem 18, prove the fundamental theorem of algebra as follows. Let p (z)be a nonconstant polynomial of degree n,p (z) = a n z n + a n−1 z n−1 + · · ·Show that for large enough r, |p (z)| > |p (z) − a n z n | for all z ∈ ∂B (0, r). Nowfrom Problem 17 you can conclude d (p, B r , 0) = d (f, B r , 0) = n where f (z) =a n z n .20. Generalize Theorem 10.7.5 to the situation where Ω is not necessarily a connectedopen set. You may need to make some adjustments on the hypotheses.21. Suppose f : R n → R n satisfies|f (x) − f (y)| ≥ α |x − y| , α > 0,Show that f must map R n onto R n . Hint: First show f is one to one. Thenuse invariance of domain. Next show, using the inequality, that the points not inf (R n ) must form an open set because if y is such a point, then there can be nosequence {f (x n )} converging to it. Finally recall that R n is connected.