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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 ; ε ) is an arbitrary neighborhood of x . We need to show this neighborhood contains an element of S . Consider the left endpoint x ε of this neighborhood. Since x ε < x , where x = sup S , then x ε is not an upper bound for S . Therefore there exists an element s ∈ S such that x ε < s ≤ x. Since x / ∈ S , s 6 = x . It then follows that s ∈ N * ( x ; ε ), 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 ; ε ) of a contained entirely in A . That is, the open interval ( a ε,a + ε ) ⊆ A . Then the Density Theorem for Q implies there exists a rational number r ∈ Q and r ∈ ( a ε,a + ε ). 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 ; ε 1 ) such that N ( x ; ε 1 ) ⊆ S . To show x ∈ int T , we need to show there exists a neighborhood N ( x ; ε 2 ) such that N ( x ; ε 2 ) ⊆ T . Since x ∈ int( S ∩ T ), then there exists a neighborhood N ( x ; ε ) such that N ( x ; ε ) ⊆ S ∩ T . Then N ( x ; ε ) ⊆ S , so x ∈ int S and N ( x ; ε ) ⊆ 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 ; ε ) such that N ( x ; ε ) ⊆ S ∩ T . Since x ∈ int S , then there exists a neighborhood N ( x ; ε 1 ) such that N ( x ; ε 1 ) ⊆ S . Since x ∈ int T , then there exists a neighborhood N ( x ; ε 2 ) such that N ( x ; ε 2 ) ⊆ T ....
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 Spring '08
 QIAN
 Calculus, Topology, Metric space, Closed set, Archimedean Property, finite subcover, S1 S2

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