- #1
Icebreaker
A and B are bounded sets. C = \{ab | a \in A, b \in B\}
Show that (Sup A)(Sup B) = Sup C.
I tried to do it as follows,
\alpha = Sup A \Rightarrow \forall \epsilon > 0, \exists a \in A s.t. \alpha - \epsilon < a
\beta= Sup B \Rightarrow \forall \epsilon > 0, \exists b \in B s.t. \beta - \epsilon < b
\alpha\beta < ab
\alpha\beta - \alpha\epsilon - \beta\epsilon + \epsilon^2 < ab
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,
\alpha = Sup A \Rightarrow \forall \epsilon > 0, \exists a \in A s.t. \alpha - \epsilon < a
\beta= Sup B \Rightarrow \forall \epsilon > 0, \exists b \in B s.t. \beta - \epsilon < b
\alpha\beta < ab
\alpha\beta - \alpha\epsilon - \beta\epsilon + \epsilon^2 < ab
No matter what I set epsilon to, I can't isolate the final epsilon. Any help?
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