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 π ; 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 deﬁnition of a limit it follows that for ε = 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 ....
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This document was uploaded on 03/27/2010.
- Spring '09