{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Section2.3Number6&8 - S ⊆ R be nonempty Show that if...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Section 2.3 #6 If S R contains one of its upper bounds, show that this upper bound is the supremum of S . Assumptions: We assume that S contains one of its upper bounds. This means that there is an element (let’s call it u ) in S such that u is an upper bound for S . Anytime you encounter a term that we have given a formal definition, you should use that definition or the theorems and lemmas we have proved using it. So since we know that u is an upper bound for S , we know that for every s S , we have s u . What we want to prove: We should prove that u is the supremum for S . Refer to the formal definition of supremum. To prove that u is a supremum for S , we must show that (1) u is an upper bound for S , and (2) if v is any upper bound for S , then v u . Since (1) is one of our assumptions, we only need to prove (2). This is an if-then statement, so assume the “if” part and prove the “then” part. Upshot: Assume (a) u is an upper bound for S and (b) u S , and suppose in addition that (c) v is an upper bound for S . You should prove v u . Section 2.3 #8
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: S ⊆ R be nonempty. Show that if u = sup S , then for every number n ∈ N the number u-1 n is not an upper bound of S , but the number u + 1 n is an upper bound of S . d To prove a “for every n ∈ N ” statement, prove it for a fixed n (but do not specify a particular example), and then say that your proof will hold for all of them, since the only assumption you used about n was that it was in N . Since upper bound and supremum are words that we have formally defined, you should use the definitions, or the theorems and lemmas that we proved using the definitions. In particular, to prove that u + 1 n is an upper bound, you must show that it satisfies a definition of upper bound. In other words, you must show that for every s ∈ S , we have s ≤ u + 1 n . Upshot: Assume u = sup S . Let n ∈ N . Use the definitions of supremum and upper bound to show that u-1 n is not an upper bound and u + 1 n is an upper bound....
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern