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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]?
     
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