Cut: A cut in (The set of rational numbers) is a set such that
If , and , then .
does not have a least element.
Real numbers: The set of real numbers is defined as the set of all cuts in .
If , are cuts, then we say that iff .
Theorem 1 is complete.
Proof: Let be bounded below.
Now consider .
We first show that is a cut.
Since as it is a cut and there exists atleast one , .
, such that .
and , and hence, .
If had a least element, say x, then since , for some . But for every , and hence such that since does not have a least element. Hence, does not have a least element. Therefore, is a cut.
Now, for every , . Hence, . Hence, . Therefore, is a lower bound for B.
Consider any other lower bound, say for B. Now, and hence .
Hence, . This implies that and . Hence, is the greatest lower bound for B.
We have shown that for any arbitrary set , there exists a greatest lower bound. Hence, for every , there exists a greatest lower bound. Hence, satisfies the greatest lower bound property. Therefore, satisfies the least upper bound property. This implies that is complete.