1. ## Cuts

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.

2. I'm not sure what the terms "lower number for a cut" means, but I'll try to answer you question anyway.

You have that X is an element of the cut $\displaystyle \xi$. There _must_ be rational numbers in $\displaystyle \xi$ larger than X, because otherwise X would be the largest number of $\displaystyle \xi$ - and by part 3) of your definition, no cut contains a greatest rational number. So $\displaystyle X_1$ is just one of those rational numbers in $\displaystyle \xi$, which are larger than X.