hw3 - MATH 444 ELEMENTARY REAL ANALYSIS HOMEWORK 3 Due date...

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MATH 444: ELEMENTARY REAL ANALYSIS HOMEWORK 3 Due date: Sep 17 (Wed) Exercises from the textbook. 2.3: 5; 8; 13 2.4: 9; 11; 13 Out-of-textbook exercises. 1. Let A, B R be nonempty sets bounded above. Prove the following: (a) sup( A B ) = sup { sup A, sup B } . (b) sup( A + B ) = sup A + sup B , where A + B := { a + b : a A, b B } . 2. For each set S below, find inf S and sup S , and prove your answers. (a) S = { 1 / n : n N } ; (b) S = { 1 /n - 1 /m : n, m N } . Hint : To prove some of your answers, you would have to use one of the corollaries of the Archimedean Property (see Section 2.4 of the textbook).
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Unformatted text preview: 3. Let S ⊆ R be nonempty and u ∈ R . Prove that u = sup S if and only if for every n ∈ N , u + 1 /n is an upper bound of S while u-1 /n is not. Hint : For the right-to-left direction, use one of the corollaries of the Archimedean Prop-erty. Also, for the same direction, in showing that u is an upper bound, you may find Exercise 2.3–8 helpful, while in showing that u is the least upper bound, you may find Lemma 2.3.4 helpful....
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  • Spring '08
  • JUNGE
  • Sets, Supremum, Archimedean Property

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