This preview shows page 1. Sign up to view the full content.
Unformatted text preview: z and w in A the path from z to 1 and then from 1 to w is contained in A . Hence A is connected. 5. a) does not converge b) converges to ; Hint: use geometry to get that that z n = 2 tan-1 n c) converges to 6. We will prove the claim by contradiction. Assume that A is closed, that for all n , z n A , that lim n z n = w , but that w 6 A . Since the complement of A is an open set (by a theorem proven in class) w is an interior point of the complement. Hence there exists r > such that B ( w,r ) A = . From the denition of a limit it follows that for = r there exists n such that for all n n it holds that | z n-w | < r . Or in other words z n B ( w,r ) . Since z n A , this implies that B ( w,r ) A 6 = . Contradiction. 7. Example: Let A = B (0 , 1) , z n = 1-1 n and w = 1 ....
View Full Document
- Spring '09