- #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?
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: