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Unformatted text preview: 2.89 Let = sup ( ), so that ( ) for every (2.43) There exists a sequence in with ( ) . Since is compact, there exists a convergent subsequence and ( ) . However, since is upper semi-continuous, ( ) lim ( ) = . Combined with (2.43), we conclude that ( ) = . 2.90 Choose some > 0. Since is uniformly continuous, there exists some > 0 such that ( ( ) , ( )) < for every , such that ( , ) < . Let ( ) be a Cauchy sequence in . There exists some such that ( , ) < for every , . Uniform continuity implies that ( ( ) , ( )) < for every , ....
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- Fall '10