Then there exists some such that .
Thus, , particularly .
So 1 must be an upper bound of .
So is a bounded set. We still do not know if there is a supremum in the rationals because , and does not carry the least upper bound property.
However, we know since the reals do have the least upper bound property that at least lies in .
Suppose 1 were not the least upper bound of .
By density of the rationals in the reals, there exists such that .
Since , then is an upper bound of .
Since , write , for . Note that , for if not then which cannot be true.
However, if this were true, then would not be an upper bound of because for , then .
So therefore cannot be an upper bound of (and certainly not its least upper bound).
Therefore, the least upper bound of is 1 as desired. QED.