Products of Suprema

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?

 

Is it possible to prove that [tex]\alpha\epsilon + \beta\epsilon - \epsilon^2 >0[/tex]?