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!

Prove inf(S)=-Sup(-S)?

  1. Sep 20, 2012 #1
    Prove inf(S)=-Sup(-S)??

    1. The problem statement, all variables and given/known data


    Let S,T be subsets of ℝ, where neither T nor S are empty and both Sup(S) and Sup(T) exist.

    Prove inf(S)=-sup(-S).

    Starting with =>

    I let x=inf(S). Then by definition, for all other lower bounds y of S, x≥y.

    I'm stuck at this point...

    Any help please?

    Thanks
     
  2. jcsd
  3. Sep 20, 2012 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Re: Prove inf(S)=-Sup(-S)??

    You have to show two things:

    • -x is an upper bound of -S
    • If y is another upper bound of -S, then [itex]-x\leq y[/itex]

    So, in order to prove that -x is an upper bound of -S. Take an element -s from -S and prove [itex]-s\leq -x[/itex]. Why is that true?
     
  4. Sep 20, 2012 #3
    Re: Prove inf(S)=-Sup(-S)??

    I'm not following along, why would -s≤-x? Unless you mean if I multiply both side by (-1), then it would be x≤s? Or am I totally off on a tangent?
     
  5. Sep 20, 2012 #4

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Re: Prove inf(S)=-Sup(-S)??

    If [itex]-s \in -S[/itex], then [itex]s \in S[/itex]. And by definition, [itex]x[/itex] is a lower bound of [itex]S[/itex]...
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Prove inf(S)=-Sup(-S)?
  1. Limit to inf problem (Replies: 2)

Loading...