hmw3s - 18.100B, Fall 2002, Solutions to Homework 3 Rudin:...

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± 18.100B, Fall 2002, Solutions to Homework 3 Rudin: (1) Chapter 2, Problem 6 Done in class on Thursday September 26. Here E X is a subset of a metric space and E ± is the set of limit points, in X, of E. (a) Prove that E ± is closed. If p X is a limit point of E ± then for each r> 0 ,B ( p, r ) E ± ± q is not empty. Since q is a limit point of E and r d ( p, q ) > 0 , B ( q, r d ( p, q )) E is an infinite set. By the triangle inequality, B ( q, r d ( p, q )) B ( p, r )so B ( p, r ) E is also infinite and p is therefore a limit point of E, i.e. p E ± . Thus E ± contains each of its limit points and it is therefore closed. (b) Prove that E and E have the same limit points. If p is a limit point of E then it is a limit point of E since E E. If p 1 is a limit point of E then B ( p, n ) ( E \{ 0 } ) decreases with n ;e itherit is infinite for all n or it is empty for large n. We show that the second 1 case cannot occur. Indeed this woould imply that B ( p, n ) ( E ± \{ p } )is infinite for all n and hence that p is a limit point of E ± ;bythe preceding result it is then a limit point of E contradicting the assumption that it is not. Thus a
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This note was uploaded on 12/16/2011 for the course STAT 5446 taught by Professor Frade during the Fall '09 term at FSU.

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hmw3s - 18.100B, Fall 2002, Solutions to Homework 3 Rudin:...

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