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test1_(takehome) - 3 Let f a,b → R be continuous Show the...

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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 ² > 0, there is a δ > 0 such that if x, ˜ x ∈ [ a,b ] with | x-˜ x | < δ , then | f ( x )-f (˜ x ) | < ² . 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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