Heine-Borel - MA250 Lecture Notes: Subsequences and the...

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Unformatted text preview: MA250 Lecture Notes: Subsequences and the Heine-Borel Theorem Manuele Santoprete February 9, 2009 1 Subsequences An useful criterion to show that a sequence does not converge is given by the following Corollary to Theorem 11.2 Corollary 1. Is ( s n ) has two subsequences that converge to different limits then ( s n ) does not converge. Example 1. Let s n = (- 1) n . The subsequence t k = (- 1) 2 k converges to 1, while v k = (- 1) 2 k +1 converges to- 1. Therefore ( s n ) does not converge. 2 Closed, Open and Compact Sets Definition 1. E is closed in R if every convergent sequence in E converges to an element of E . Example 2. [ a, b ] is closed in R (see below for a proof), R is closed, the empty set is closed. Example 3. The half-open interval (0 , 1] is not closed. In fact let s n = 1 n , then s n (0 , 1] but lim s n = 0 / (0 , 1]. Definition 2. A set E is open if it is the complement of a closed set, that is E = R \ U , where U is closed. Example 4. ( a, b ) is open. 1 Example 5. = R \ R . Since R is closed it follows that is open. Similarly R = R \ . Hence, since is closed it follows that R is open. Therefore R and are both open and closed....
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Heine-Borel - MA250 Lecture Notes: Subsequences and the...

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