Prove that if A has a least upper bound, then it is unique. Thus prove: If b and b' are both least upperbound of A, then b = b'.
In general if $\displaystyle t=\text{lub}(A)$ and $\displaystyle s<t$ then this must be true $\displaystyle \left( {\exists x \in A} \right)\left[ {s < x \leqslant t} \right]$
If $\displaystyle b\ne b'$ then $\displaystyle b<b'\text{ or }b'<b$.
There are contradictions either way.