Thread: Cuts

1. Cuts

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

Theorem 129:
I) Let $\xi$ and $\eta$ be cuts. Then the set of all rational numbers which are representable in the form $X+Y$, where $X$ is a lower number for $\xi$ and $Y$ is a lower number for $\eta$, is itself a cut.

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

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 $X+Y$ 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 $X+Y$ where $X$ and $Y$ are lower numbers for $\xi$ and $\eta$ respectively. Using the third property of cuts (see * below) as applied to $\xi$, we can find a lower number
$x_1>x$
for $\xi$; then
$x_1+y>X+y$
So that there exists in our set a number which is $>X+Y$.

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

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