Prove that if a is an upper bound for A and if a is also an

Prove that if a is an upper bound for A, and if a is also an element of A, then it must be that a = sup A.

Solution

If a is an upper bound for A, then sup(A) exists, and:

sup(A) <= a ......(1)

If a is an element of A, then, since sup(A) is an upper bound on A, we have:

a <= sup(A) ........(2)

From (1) and (2) we get

Thus, sup(A) = a.