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# CompactRN - Econ 4111 Professor John Nachbar Compactness...

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Econ 4111 Professor: John Nachbar 9/25/08 Compactness and Completeness in R N . 1 R is complete. Theorem 6, the Heine-Borel 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 ε > 0, there is a T such that for all t > T , x t < b + ε . Proof. By contraposition. Consider any x such that x t x + ε for infinitely many t . Then b t x + ε for all t , hence b x + ε , hence b > x , and in particular, b 6 = x . 2. Claim. For any ε > 0, x t > b - ε for infinitely many t . Proof. By contra- position. Consider any x such that for any ε > 0 there is a T such that, for all t > T , x t x - ε . Then for all such t , b t x - ε , hence b x - ε , 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 lim sup of { x t } . 1

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From these two claims it follows that for any ε > 0, there are infinitely many t such that x t N ε ( 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, x t k converges to b .
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CompactRN - Econ 4111 Professor John Nachbar Compactness...

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