The following is from "Foundation of Analysis," by Edmund Landau:

Theorem 129:

I) Let $\displaystyle \xi$ and $\displaystyle \eta$ be cuts. Then the set of all rational numbers which are representable in the form $\displaystyle X+Y$, where $\displaystyle X$ is a lower number for $\displaystyle \xi$ and $\displaystyle Y$ is a lower number for $\displaystyle \eta$, is itself a cut.

II) No number of this set can be written as a sum of an upper number for $\displaystyle \xi$ and an upper number for $\displaystyle \eta$

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 $\displaystyle X+Y$ 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 $\displaystyle X+Y$ where $\displaystyle X$ and $\displaystyle Y$ are lower numbers for $\displaystyle \xi$ and $\displaystyle \eta$ respectively. Using the third property of cuts (see * below) as applied to $\displaystyle \xi$,we can find a lower number

$\displaystyle x_1>x$

for $\displaystyle \xi$; then

$\displaystyle x_1+y>X+y$

So that there exists in our set a number which is $\displaystyle >X+Y$.

Question: The author said, "we can find a lower number $\displaystyle X_1>X$ for $\displaystyle \xi$."? Is $\displaystyle X_1$ the lower number? Is $\displaystyle X_1$ in $\displaystyle \xi$?

*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.