The following is from "Foundation of Analysis," by Edmund Landau:
I) Let and be cuts. Then the set of all rational numbers which are representable in the form , where is a lower number for and is a lower number for , is itself a cut.
II) No number of this set can be written as a sum of an upper number for and an upper number for
The author provided a 3-step proof, in which step 1, he proved the validity of statement (II). I understood this part.
In step 2, he proved is a cut. I also understood this part, but I don't understand step 3.
Step 3: Any given number of our set is the form where and are lower numbers for and respectively. Using the third property of cuts (see * below) as applied to , we can find a lower number
for ; then
So that there exists in our set a number which is .
Question: The author said, "we can find a lower number for ."? Is the lower number? Is in ?
*Definition 28: A set of rational numbers is called a cut if
1) it contains a rational number, but does not contain all rational numbers;
2) every rational number of the set is smaller than every rational number not belonging to the set;
3) it does not contain a greatest rational number.