1.9. CONVERGENCE IN C 21 The closed ball of radius r with center z0 is (1.2) ¯ r (z0) = {z C | |z z0| r}. The open balls and closed balls in C are the analogs of open and closed intervals in R. We can define open and closed sets in C in a fashion similar to the definitions in R. Definition 1.9.3. Let S be a subset of C. We say that S is an open set in C if, for each point z S, there is an ε 0 (depending on z) such that Bε(z) S. Definition 1.9.4. Let S be a subset of C. We say that S is a closed set in C if the complement of S is an open set in C. Note that the empty set and C are both open and closed subsets of C. Exercise 1.9.5. (i) Show that and C are the only subsets of C which are both open and closed in C. (ii) Show that every open set in C can be written as a countable union of open balls. (iii) Show, by example, that there are open sets in C for which the open balls in (ii) cannot be made pairwise disjoint. (iv) Show that an arbitrary union of open sets in C is an open set in C. (v) Show that a finite intersection of open sets in C is an open set in C. (vi) Show, by example, that an infinite intersection of open sets in C need not be an open set in C. (vii) Show that an arbitrary intersection of closed sets in C is a closed set in C. (viii) Show that a finite union of closed sets in C is a closed set in C. (ix) Show, by example, that an infinite union of closed sets in C is not necessarily a closed set in C. Exercise 1.9.6. Consider the collection of open balls {Br(z)} in C where r Q and where Re z and Im z Q. Show that any open set in C can be written as a finite or countable union from this collection of sets. Definition 1.9.7. Let A C. The set A is bounded if there exists r 0 such that A Br(0). Exercise 1.9.8. Define the notion of a bounded sequence in C. Definition 1.9.9 (See Definition 1.6.12). A sequence (zk)k∈N in C is a Cauchy sequence if, given any ε 0, there exists N N such that n, m N implies |zn zm| ε. Exercise 1.9.10. Prove that every Cauchy sequence in C is bounded. Theorem 1.9.11 (Cauchy criterion). A sequence (zk)k∈N of complex numbers is convergent if and only if it is a Cauchy sequence.
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