This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Econ 4111 Professor: John Nachbar 9/25/08 Compactness and Completeness in R N . 1 R is complete. Theorem 6, the HeineBorel theorem, states that a set in R N is compact iff it is closed and bounded. Theorem 6 is immediate if I can show (a) that R N is complete and (b) that bounded sets in R N are totally bounded. I demonstrate the latter in Theorem 5. As for completeness, I start by showing here that R is complete, then show that this implies that R N is complete. Theorem 1. R is complete. Proof. Let { x t } be a Cauchy sequence in R . I claim that { x t } is bounded. Since { x t } is Cauchy, there is a T such that for any s > T , d ( x T +1 ,x s ) < 1. Set r = max { d ( x 1 ,x T +1 ) ,...,d ( x T ,x T +1 ) } + 1 . Then for any t , d ( x t ,x T +1 ) < r , as was to be shown. Therefore, for each t , the set { x t ,x t +1 ,... } is bounded above and hence has a supremum b t . Moreover, the set { x 1 ,x 2 ,... } is bounded below and hence has an infimum. Therefore, for every t , b t inf { x 1 ,x 2 ,... } . Therefore, the set { b 1 ,b 2 ,... } is bounded below and so has an infimum. Call this infimum b . 1 I claim that { x t } has a subsequence converging to b . This follows from two properties of b . 1. Claim. For any > 0, there is a T such that for all t > T , x t < b + . Proof. By contraposition. Consider any x such that x t x + for infinitely many t . Then b t x + for all t , hence b x + , hence b > x , and in particular, b 6 = x . 2. Claim. For any > 0, x t > b for infinitely many t . Proof. By contra position. Consider any x such that for any > 0 there is a T such that, for all t > T , x t x . Then for all such t , b t x , hence b x , hence b < x and, in particular, b 6 = x . 1 For example, suppose that x t = 1 + 1 /t if t is odd and x t = 1 /t if t is even. Then for each t , a t = 1 /t , b t = 1 + 1 /t and b = 1. The point b is called the limsup of { x t } . 1 From these two claims it follows that for any > 0, there are infinitely many t such that x t N ( b ). Choose t 1 such that x t 1 N 1 ( b ), t 2 > t 1 such that x t 2 N 1 / 2 ( b ), and so on. By construction,and so on....
View
Full
Document
This note was uploaded on 05/20/2010 for the course ECON 511 taught by Professor Johnnachbar during the Fall '08 term at Washington University in St. Louis.
 Fall '08
 JohnNachbar

Click to edit the document details