Is Every Element Greater Than U Not in Set S?

[Punkyc7]

Let S$\subseteq$ R be non empty. Show that u$\in$R is an upperbound of S iff the conditions t $\in$R and t>u implies t$\notin$S.

Let S$\subseteq$ R be non empty. Assume u$\in$R is an upperbound of S. Then for all x$\in$S x$\leq$u. Then choose a t$\in$R such that t>u. Since t>u this implies that t$\notin$S since u=SupS

Let S$\subseteq$ R be non empty and t $\in$R and t>u implies t$\notin$S for some u $\in$R. So either u$\in$S or u$\notin$S. If u$\notin$S then u is an upper bound of S. So consider u$\in$S and let u be the largest element is S such that u<t. This implies that us us the largest element in S since t$\notin$S.
Is this right?

 

[mighty2000]

Yes, your response is correct. You have effectively shown that u is an upperbound of S if and only if t \inR and t>u implies t\notinS for all t \inR. Well done!