HW2 - C such that their intersection is not connected 4...

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P ROBLEM SET 2 due in class on Friday, January 23. Complex Variables MAT334 Remark: Please try to write clearly. In particular every statement that you make in your solution should follow from the assumptions of the problem, some of the material that was a prerequisite for the course (high-school algebra, calculus), was done in class, or some of the statements that you have already established in your solution. If you are using some of the theorems that have a name, mention that (e.g. ”By Mean Value Theorem. ..”). 1. Show that for A C the following holds: a) ( ∂A ) ∂A and b) ( A ∂A ) ∂A . Note that these facts imply that ∂A and A ∂A are closed sets. The set A ∂A is usually denoted by A and called the the closure of A . 2. Show that A is open if and only if A ∂A = . 3. Give an example of two open, connected sets in
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Unformatted text preview: C such that their intersection is not connected. 4. Show that the set C \{ z | z ∈ R and z ≤ } (that is the set of complex numbers without negative real numbers) is open and connected. 5. Find the limit of each sequence that converges; if the sequence diverges, explain why. a) z n = ± 1 + i √ 2 ² n , b) z n = Arg ((1 + ni ) 2 ) , c) z n = n 2 ± i 2 ² n 6. Let A be a closed set and lim n →∞ z n = w . Show that if z n ∈ A for all n then w ∈ A . 7. Show that the statement of the previous problem would not be true in general if A was an open set. In other words present an open set A and a convergent sequence, such that all the elements of the sequence are in A but the limit is not in A ....
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