Suppose that b is an upper bound for a set S of real numbers. Prove that if b E S, then b = lub S.
$\displaystyle S\leq b$ $\displaystyle \forall$ $\displaystyle s\in S.$
Assume $\displaystyle b \in S.$
Set $\displaystyle m=\sup{S}$ so, $\displaystyle S\leq b \leq m$ by assumption.
Then, since $\displaystyle b$ is an upper bound, and $\displaystyle m$ is the $\displaystyle l.u.b.$ $\displaystyle S\leq m \leq b.$
Therefore $\displaystyle b \in S \Rightarrow b= m =\sup{S}$