Suppose that F is an ordered field and that A subsetofequalt

Suppose that F is an ordered field and that A subsetofequalto F is a subset which has a least upper bound. Then show that if s and s\' are least upper bounds for A, then s = s\'. (That is, the least upper bound of a set is unique.)

Solution

Sol. Let s and s\' be two least upper bounds for A.

Since s is least upper bound and s\' upper bound of A

Then s<=s\' ...............(1)

Now s\' is least upper bound and s is upper bound of A

Then s\'<=s..........(2)

From (1) & (2) we get

S=S\'