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Unformatted text preview: Math 350 Advanced Calculus I Spring 2008 Homework Solutions 4 # 13: Textbook Chapter 13 13.14 Let S be a bounded infinite set and let x = sup S . Prove: If x / S , then x S . Solution Suppose x / S . We need to show x S . That is, we need to show x is an accumulation point of S . To show this, we need to show that every deleted neighborhood of x contains an element of S . Suppose N ( x ; ) is an arbitrary neighborhood of x . We need to show this neighborhood contains an element of S . Consider the left endpoint x of this neighborhood. Since x < x , where x = sup S , then x is not an upper bound for S . Therefore there exists an element s S such that x < s x. Since x / S , s 6 = x . It then follows that s N * ( x ; ), as required. Thus x S . 13.19 Let A be a nonempty open subset of R and let Q be the set of rational numbers. Prove that A Q 6 = . Solution To prove that A Q 6 = , we need to show that the set A must contain a rational number. Since A is nonempty, then we can choose an element a A . Since A is open, then by Theorem 13.7, A = int A , hence a int A . Therefore there exists a neighborhood N ( a ; ) of a contained entirely in A . That is, the open interval ( a ,a + ) A . Then the Density Theorem for Q implies there exists a rational number r Q and r ( a ,a + ). Since this interval is contained in A , then r A . Thus the set A contains a rational number. 13.21 Let S and T be subsets of R. Prove the following: (c) int( S T ) = (int S ) (int T ) Solution Forward Direction. Suppose x is an arbitrary real number. Suppose x int( S T ) . We need to show x (int S ) (int T ) . To show this, we need to show x int S and x int T. To show x int S , we need to show there exists a neighborhood N ( x ; 1 ) such that N ( x ; 1 ) S . To show x int T , we need to show there exists a neighborhood N ( x ; 2 ) such that N ( x ; 2 ) T . Since x int( S T ), then there exists a neighborhood N ( x ; ) such that N ( x ; ) S T . Then N ( x ; ) S , so x int S and N ( x ; ) T , so x int T . Therefore x (int S ) (int T ) . This proves the forward direction. 1 Backward Direction. Suppose x is an arbitrary real number. Suppose x (int S ) (int T ) . We need to show x int( S T ) . To show this, we need to show there exists a neighborhood N ( x ; ) such that N ( x ; ) S T . Since x int S , then there exists a neighborhood N ( x ; 1 ) such that N ( x ; 1 ) S . Since x int T , then there exists a neighborhood N ( x ; 2 ) such that N ( x ; 2 ) T ....
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 Spring '08
 QIAN
 Calculus

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