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!

Products of Suprema

  1. Sep 25, 2005 #1
    A and B are bounded sets. [tex]C = \{ab | a \in A, b \in B\}[/tex]
    Show that (Sup A)(Sup B) = Sup C.

    I tried to do it as follows,

    [tex]\alpha = Sup A \Rightarrow \forall \epsilon > 0, \exists a \in A s.t. \alpha - \epsilon < a[/tex]

    [tex]\beta= Sup B \Rightarrow \forall \epsilon > 0, \exists b \in B s.t. \beta - \epsilon < b[/tex]

    [tex]\alpha\beta < ab[/tex]

    [tex]\alpha\beta - \alpha\epsilon - \beta\epsilon + \epsilon^2 < ab[/tex]

    No matter what I set epsilon to, I can't isolate the final epsilon. Any help?
    Last edited by a moderator: Sep 25, 2005
  2. jcsd
  3. Sep 25, 2005 #2
    Is it possible to prove that [tex]\alpha\epsilon + \beta\epsilon - \epsilon^2 >0[/tex]?
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Products Suprema Date
Algorithm to matrix product MSR format Nov 22, 2017
Dot product? Jun 10, 2017
Pair production problem May 18, 2017
Suprema and Infima Jan 25, 2005