test1_(takehome) - 3: Let f : [ a,b ] R be continuous. Show...

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(Take Home part) Test 1, MAA4226, Due Sep 29th Student Name: It is considered cheating to collaborate on these problems. Write each solution on a separate sheet, arrange them in increasing order, then put this page on top and staple. 1: Show that the completeness axiom can be replaced by the following statement Each monotone decreasing sequence which is bounded from below has a limit . In other words, assume the statement above and prove that For each set A R bounded from above there exists a least upper bound, the sup ( A ) . 2: Let { a n } be a sequence of positives such that lim n →∞ a n +1 a n = l. a)Show that lim n →∞ a n = l . Hint: Use the Mean Inequality b) Find an example to show the converse is false.
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Unformatted text preview: 3: Let f : [ a,b ] R be continuous. Show the followings: a) For any > 0, there is a > 0 such that if x, x [ a,b ] with | x- x | < , then | f ( x )-f ( x ) | < . b) Give an example of continuous f : (0 , 1) R for which the statement in part (a) fails. Justify your answer. 4: Let f : (0 , 1) R be increasing and bounded. Show that lim x f ( x ) exists. Important results: Archimedean property, monotone and bounded sequences are conver-gent, closed and bounded sets are compact, Extreme Value Theorem, Intermediate Value Theorem, continuous maps on compacta are uniformly continuous. 1...
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This note was uploaded on 11/18/2011 for the course MAP 4426 taught by Professor Tamasan during the Fall '10 term at University of Central Florida.

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