Solutions to HW 4

Solutions to HW 4 - Math 350 Advanced Calculus I Spring...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 350 Advanced Calculus I Spring 2008 Homework Solutions 4 # 1-3: 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 ....
View Full Document

Page1 / 9

Solutions to HW 4 - Math 350 Advanced Calculus I Spring...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online