1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

How to prove a number is a supremum of a set

  1. Oct 4, 2009 #1
    1. The problem statement, all variables and given/known data
    Prove that if
    (i) [tex]\forall n[/tex][tex]\in N[/tex], u - (1/n) is not an upper bound of s
    (ii) [tex]\forall n[/tex][tex]\in N[/tex], u + (1/n) is an upper bound of S
    then, u = supS

    2. Relevant equations



    3. The attempt at a solution
    It (i) and (ii) are true, then
    [tex]\exists s[/tex][tex]\in S[/tex] s.t. u - (1/n) < s
    and u+(1/n)>s for all s.
    I'm not sure where to go from here.
     
  2. jcsd
  3. Oct 4, 2009 #2

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Re: suprenums

    How about a proof by contradiction?

    Suppose [itex]u > \sup S[/itex]. Then there exists [itex]n \in \mathbb{N}[/itex] such that [itex]\sup S < u - 1/n < u[/itex]. Does this violate one of (i) or (ii)?

    Next suppose [itex]u < \sup S[/itex]. Can you rule this out as well?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: How to prove a number is a supremum of a set
  1. Supremum of sets (Replies: 3)

Loading...