This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 350 Advanced Calculus I Fall 2009 Homework Solutions Section 12 THE COMPLETENESS AXIOM 12.3 For each of the following subsets of R , give its supremum and its maximum, if they exist. For each part, you must verify your results. That is, if you are claiming that a number a is the supremum, you must show that it is using the definition. (d) S = (0 , 4) sup S = 4, No maximum We want to prove that sup S = 4. To show this, we need to show (1) 4 is an upper bound for S . To show this, we need to show ( s S ) [ s 4] . Suppose s is an arbitrary element of S . Then 0 < s < 4, so s 4, as required. Therefore 4 is an upper bound for S . (2) For all m R , if m is an upper bound for S , then 4 m . To prove this, we use a proof by contrapositive. The contrapositive statement is ( m R ) [ m < 4 ( m is not an upper bound for S )] . Suppose m is an arbitrary real number and m < 4. We need to show that m is not an upper bound for S . To show this, we need to show there exists an element s S such that s > m . Find an s . In the case that m 0, then every element of S is greater than m , so choose s = 1, for instance. In the case that m > 0, choose s to be the midpoint of m and 4, so s = 1 2 ( m + 4) . Verify s satisfies: (i) Domain: s S (ii) Propositional Function: s > m In the case that m 0, then s = 1 S and s = 1 > m . In the case that m > 0, then m S . Since s = 1 2 ( m + 4) is the midpoint of m and 4, then s S also and s > m , as required. Therefore m is not an upper bound for S . This proves the contrapositive statement. Thus 4 is the least upper bound of S . To show that S has no maximum, note that since sup S = 4, then no number less than 4 can be an upper bound of S . Since all elements of S are less than 4, then it follows that S does not contain any upper bounds so S has no maximum....
View
Full
Document
 Spring '08
 QIAN
 Calculus, Sets

Click to edit the document details