I would do it this way:

Suppose x in A is an upper bound for A, but that x is not the supremum of A.

Then there exists another upper bound y for A, such that y<x. But this conradicts y being an upper bound for A, as there is an element x of A such that x>y.

Hence if x in A, is an upper bound for A a subset of R, then x = sup(A).

RonL